answer：Okay, so I have this problem where I need to find the volume of a solid formed by revolving a region S around a line. The region S is defined by two inequalities: |6 - x| + y ≤ 8 and 4y - x ≥ 20. The line around which we're revolving is 4y - x = 20. Hmm, okay, let me try to break this down step by step.First, I need to understand the region S. It's defined by two inequalities, so I should probably graph them to visualize what S looks like. The first inequality is |6 - x| + y ≤ 8. That seems like an absolute value inequality, which usually forms a V-shaped graph. Let me rewrite it to make it clearer: y ≤ 8 - |x - 6|. So, this is a V-shaped graph opening downward with its vertex at (6, 8). The arms of the V will have slopes of 1 and -1 because the coefficient of |x - 6| is 1.The second inequality is 4y - x ≥ 20. Let me rewrite that as y ≥ (x + 20)/4. This is a linear inequality, so it's a straight line with a slope of 1/4 and a y-intercept at 5. Since it's a "greater than or equal to," the region we're interested in is above this line.So, region S is the area where both these inequalities are satisfied. That means it's the area above the line y = (x + 20)/4 and below the V-shaped graph y = 8 - |x - 6|. To find the exact boundaries, I need to find where these two graphs intersect.Let me set them equal to each other to find the points of intersection:(x + 20)/4 = 8 - |x - 6|This equation will have two cases because of the absolute value. Let me consider both cases.Case 1: x - 6 is non-negative, so |x - 6| = x - 6.Then the equation becomes:(x + 20)/4 = 8 - (x - 6)Simplify the right side:8 - x + 6 = 14 - xSo, (x + 20)/4 = 14 - xMultiply both sides by 4 to eliminate the denominator:x + 20 = 56 - 4xBring all terms to one side:x + 4x + 20 - 56 = 05x - 36 = 05x = 36x = 36/5 = 7.2Now, let's find the corresponding y value:y = (7.2 + 20)/4 = (27.2)/4 = 6.8So, one point of intersection is (7.2, 6.8).Case 2: x - 6 is negative, so |x - 6| = -(x - 6) = 6 - x.Then the equation becomes:(x + 20)/4 = 8 - (6 - x)Simplify the right side:8 - 6 + x = 2 + xSo, (x + 20)/4 = x + 2Multiply both sides by 4:x + 20 = 4x + 8Bring all terms to one side:x - 4x + 20 - 8 = 0-3x + 12 = 0-3x = -12x = 4Find the corresponding y value:y = (4 + 20)/4 = 24/4 = 6So, the other point of intersection is (4, 6).Wait, hold on. When I set x = 4, is |x - 6| equal to 6 - x? Yes, because 4 - 6 is negative. So, that's correct.So, the region S is bounded between x = 4 and x = 7.2, above the line y = (x + 20)/4 and below the V-shaped graph y = 8 - |x - 6|.Now, I need to revolve this region around the line 4y - x = 20. Hmm, that line is the same as the second inequality, which is y = (x + 20)/4. So, we're revolving the region around the line that forms its lower boundary.This sounds like a problem where I can use the method of disks or washers. But since we're revolving around a slant line, it might be a bit more complicated. Maybe I should consider using the method of cylindrical shells or perhaps a coordinate transformation to make the axis of revolution align with one of the coordinate axes.Wait, another thought: since the axis of revolution is a straight line, maybe I can perform a coordinate transformation to rotate the coordinate system so that the axis of revolution becomes one of the coordinate axes, say the x-axis or y-axis. That might simplify the problem.Let me think about how to do that. The line 4y - x = 20 can be rewritten as y = (x + 20)/4, which has a slope of 1/4. To align this with, say, the x-axis, I would need to rotate the coordinate system by an angle θ where tanθ = 1/4. That way, the line becomes the new x-axis in the rotated system.But this might be a bit involved. Maybe there's a simpler way. Alternatively, I can use the method of Pappus's Centroid Theorem, which states that the volume of a solid obtained by rotating a plane figure about an external axis is equal to the product of the area of the figure and the distance traveled by its centroid.So, if I can find the area of region S and the centroid of S, then the volume would be Area × (distance traveled by centroid). The distance traveled by the centroid would be the circumference of the circle it makes around the axis, which is 2π times the distance from the centroid to the axis.This might be a good approach because it avoids dealing with the complexities of integrating around a slant line.Alright, let's try that.First, I need to find the area of region S. Since S is bounded by two curves between x = 4 and x = 7.2, I can set up an integral to find the area.The upper boundary is y = 8 - |x - 6|, and the lower boundary is y = (x + 20)/4.So, the area A is the integral from x = 4 to x = 7.2 of [upper function - lower function] dx.But wait, the upper function is a V-shaped graph, so it's actually two different linear functions depending on whether x is less than 6 or greater than 6.So, I need to split the integral into two parts: from x = 4 to x = 6, and from x = 6 to x = 7.2.From x = 4 to x = 6, |x - 6| = 6 - x, so the upper function is y = 8 - (6 - x) = 2 + x.From x = 6 to x = 7.2, |x - 6| = x - 6, so the upper function is y = 8 - (x - 6) = 14 - x.Therefore, the area A is:A = ∫ from 4 to 6 [ (2 + x) - ( (x + 20)/4 ) ] dx + ∫ from 6 to 7.2 [ (14 - x) - ( (x + 20)/4 ) ] dxLet me compute each integral separately.First integral, from 4 to 6:(2 + x) - (x + 20)/4 = 2 + x - x/4 - 5 = (2 - 5) + (x - x/4) = (-3) + (3x/4) = (3x/4) - 3So, integral becomes ∫ from 4 to 6 (3x/4 - 3) dxIntegrate term by term:∫ (3x/4) dx = (3/8)x²∫ (-3) dx = -3xSo, the first integral is [ (3/8)x² - 3x ] evaluated from 4 to 6.Compute at x = 6:(3/8)(36) - 3*6 = (108/8) - 18 = 13.5 - 18 = -4.5Compute at x = 4:(3/8)(16) - 3*4 = (48/8) - 12 = 6 - 12 = -6So, the first integral is (-4.5) - (-6) = 1.5Second integral, from 6 to 7.2:(14 - x) - (x + 20)/4 = 14 - x - x/4 - 5 = (14 - 5) + (-x - x/4) = 9 - (5x/4)So, integral becomes ∫ from 6 to 7.2 (9 - (5x/4)) dxIntegrate term by term:∫ 9 dx = 9x∫ (5x/4) dx = (5/8)x²So, the second integral is [9x - (5/8)x²] evaluated from 6 to 7.2Compute at x = 7.2:9*(7.2) - (5/8)*(7.2)²First, 9*7.2 = 64.8Second, (7.2)² = 51.84So, (5/8)*51.84 = (5*51.84)/8 = 259.2/8 = 32.4So, 64.8 - 32.4 = 32.4Compute at x = 6:9*6 - (5/8)*(36) = 54 - (180/8) = 54 - 22.5 = 31.5So, the second integral is 32.4 - 31.5 = 0.9Therefore, total area A = 1.5 + 0.9 = 2.4Wait, that seems a bit small. Let me double-check my calculations.First integral:From 4 to 6, (3x/4 - 3) dxAt x = 6: (3/8)*(36) - 3*6 = 13.5 - 18 = -4.5At x = 4: (3/8)*(16) - 3*4 = 6 - 12 = -6Difference: -4.5 - (-6) = 1.5. That seems correct.Second integral:From 6 to 7.2, (9 - 5x/4) dxAt x = 7.2: 9*7.2 = 64.8; (5/8)*(7.2)^2 = (5/8)*51.84 = 32.4; so 64.8 - 32.4 = 32.4At x = 6: 9*6 = 54; (5/8)*(36) = 22.5; so 54 - 22.5 = 31.5Difference: 32.4 - 31.5 = 0.9. That also seems correct.So, total area is indeed 1.5 + 0.9 = 2.4. Hmm, okay.Now, I need to find the centroid of region S. The centroid (x̄, ȳ) is given by the formulas:x̄ = (1/A) ∫ x dAȳ = (1/A) ∫ y dABut since we're dealing with a region between two curves, we can express these integrals as:x̄ = (1/A) [ ∫ from 4 to 6 ∫ from (x+20)/4 to (2 + x) x dy dx + ∫ from 6 to 7.2 ∫ from (x+20)/4 to (14 - x) x dy dx ]Similarly for ȳ:ȳ = (1/A) [ ∫ from 4 to 6 ∫ from (x+20)/4 to (2 + x) y dy dx + ∫ from 6 to 7.2 ∫ from (x+20)/4 to (14 - x) y dy dx ]This might get a bit involved, but let's proceed step by step.First, let's compute x̄.Compute the first part: ∫ from 4 to 6 ∫ from (x+20)/4 to (2 + x) x dy dxThe inner integral with respect to y is x*(upper - lower) = x*( (2 + x) - (x + 20)/4 )We already computed this earlier as (3x/4 - 3). So, the integral becomes ∫ from 4 to 6 x*(3x/4 - 3) dxWhich is ∫ from 4 to 6 (3x²/4 - 3x) dxIntegrate term by term:∫ (3x²/4) dx = (3/12)x³ = (1/4)x³∫ (-3x) dx = (-3/2)x²So, the integral is [ (1/4)x³ - (3/2)x² ] evaluated from 4 to 6.Compute at x = 6:(1/4)*(216) - (3/2)*(36) = 54 - 54 = 0Compute at x = 4:(1/4)*(64) - (3/2)*(16) = 16 - 24 = -8So, the first part is 0 - (-8) = 8Second part: ∫ from 6 to 7.2 ∫ from (x+20)/4 to (14 - x) x dy dxAgain, the inner integral is x*(upper - lower) = x*( (14 - x) - (x + 20)/4 )Compute (14 - x) - (x + 20)/4:14 - x - x/4 - 5 = (14 - 5) + (-x - x/4) = 9 - (5x/4)So, the integral becomes ∫ from 6 to 7.2 x*(9 - 5x/4) dxWhich is ∫ from 6 to 7.2 (9x - (5x²)/4) dxIntegrate term by term:∫ 9x dx = (9/2)x²∫ (5x²)/4 dx = (5/12)x³So, the integral is [ (9/2)x² - (5/12)x³ ] evaluated from 6 to 7.2Compute at x = 7.2:(9/2)*(7.2)^2 - (5/12)*(7.2)^3First, (7.2)^2 = 51.84; (7.2)^3 = 373.248So,(9/2)*51.84 = (9*51.84)/2 = 466.56/2 = 233.28(5/12)*373.248 = (5*373.248)/12 ≈ 1866.24/12 ≈ 155.52So, 233.28 - 155.52 = 77.76Compute at x = 6:(9/2)*(36) - (5/12)*(216) = (9*36)/2 - (5*216)/12 = 162 - 90 = 72So, the second part is 77.76 - 72 = 5.76Therefore, total x̄ integral is 8 + 5.76 = 13.76So, x̄ = 13.76 / 2.4 ≈ 5.7333Hmm, 13.76 divided by 2.4. Let me compute that:2.4 * 5 = 122.4 * 5.7 = 13.68So, 13.76 - 13.68 = 0.08So, 0.08 / 2.4 ≈ 0.0333So, x̄ ≈ 5.7 + 0.0333 ≈ 5.7333Alternatively, 13.76 / 2.4 = (13.76 * 10) / 24 = 137.6 / 24 ≈ 5.7333So, x̄ ≈ 5.7333Now, let's compute ȳ.Similarly, ȳ = (1/A) [ ∫ from 4 to 6 ∫ from (x+20)/4 to (2 + x) y dy dx + ∫ from 6 to 7.2 ∫ from (x+20)/4 to (14 - x) y dy dx ]First part: ∫ from 4 to 6 ∫ from (x+20)/4 to (2 + x) y dy dxThe inner integral with respect to y is (1/2)*(upper² - lower²)So, upper = 2 + x, lower = (x + 20)/4So, inner integral = (1/2)*[ (2 + x)^2 - ( (x + 20)/4 )^2 ]Compute this:(2 + x)^2 = 4 + 4x + x²( (x + 20)/4 )^2 = (x² + 40x + 400)/16So, inner integral = (1/2)*[ (4 + 4x + x²) - (x² + 40x + 400)/16 ]Let me compute this step by step.First, expand the subtraction:= (1/2)*[ (4 + 4x + x²) - (x² + 40x + 400)/16 ]To combine these, I'll get a common denominator of 16:= (1/2)*[ (64 + 64x + 16x² - x² - 40x - 400)/16 ]Simplify numerator:64 - 400 = -33664x - 40x = 24x16x² - x² = 15x²So, numerator = 15x² + 24x - 336Thus, inner integral = (1/2)*(15x² + 24x - 336)/16 = (15x² + 24x - 336)/32So, the first part integral becomes ∫ from 4 to 6 (15x² + 24x - 336)/32 dxFactor out 1/32:= (1/32) ∫ from 4 to 6 (15x² + 24x - 336) dxIntegrate term by term:∫ 15x² dx = 5x³∫ 24x dx = 12x²∫ (-336) dx = -336xSo, the integral is (1/32)[5x³ + 12x² - 336x] evaluated from 4 to 6.Compute at x = 6:5*(216) + 12*(36) - 336*6 = 1080 + 432 - 2016 = (1080 + 432) - 2016 = 1512 - 2016 = -504Compute at x = 4:5*(64) + 12*(16) - 336*4 = 320 + 192 - 1344 = (320 + 192) - 1344 = 512 - 1344 = -832So, the integral is (1/32)[(-504) - (-832)] = (1/32)(328) = 10.25Second part: ∫ from 6 to 7.2 ∫ from (x+20)/4 to (14 - x) y dy dxAgain, inner integral is (1/2)*(upper² - lower²)Upper = 14 - x, lower = (x + 20)/4So, inner integral = (1/2)*[ (14 - x)^2 - ( (x + 20)/4 )^2 ]Compute this:(14 - x)^2 = 196 - 28x + x²( (x + 20)/4 )^2 = (x² + 40x + 400)/16So, inner integral = (1/2)*[ (196 - 28x + x²) - (x² + 40x + 400)/16 ]Again, get a common denominator of 16:= (1/2)*[ (3136 - 448x + 16x² - x² - 40x - 400)/16 ]Simplify numerator:3136 - 400 = 2736-448x - 40x = -488x16x² - x² = 15x²So, numerator = 15x² - 488x + 2736Thus, inner integral = (1/2)*(15x² - 488x + 2736)/16 = (15x² - 488x + 2736)/32So, the second part integral becomes ∫ from 6 to 7.2 (15x² - 488x + 2736)/32 dxFactor out 1/32:= (1/32) ∫ from 6 to 7.2 (15x² - 488x + 2736) dxIntegrate term by term:∫ 15x² dx = 5x³∫ (-488x) dx = -244x²∫ 2736 dx = 2736xSo, the integral is (1/32)[5x³ - 244x² + 2736x] evaluated from 6 to 7.2Compute at x = 7.2:5*(7.2)^3 - 244*(7.2)^2 + 2736*(7.2)First, (7.2)^2 = 51.84; (7.2)^3 = 373.248So,5*373.248 = 1866.24244*51.84 = let's compute 244*50 = 12,200 and 244*1.84 ≈ 244*1.8 = 439.2 and 244*0.04=9.76, so total ≈ 439.2 + 9.76 ≈ 448.96, so total 244*51.84 ≈ 12,200 + 448.96 ≈ 12,648.962736*7.2 = let's compute 2736*7 = 19,152 and 2736*0.2 = 547.2, so total ≈ 19,152 + 547.2 = 19,699.2So, putting it all together:1866.24 - 12,648.96 + 19,699.2 ≈ (1866.24 + 19,699.2) - 12,648.96 ≈ 21,565.44 - 12,648.96 ≈ 8,916.48Compute at x = 6:5*(216) - 244*(36) + 2736*65*216 = 1080244*36 = 8,7842736*6 = 16,416So,1080 - 8,784 + 16,416 = (1080 + 16,416) - 8,784 = 17,496 - 8,784 = 8,712So, the integral is (1/32)(8,916.48 - 8,712) = (1/32)(204.48) = 6.39Therefore, total ȳ integral is 10.25 + 6.39 = 16.64So, ȳ = 16.64 / 2.4 ≈ 6.9333Wait, 16.64 divided by 2.4. Let me compute that:2.4 * 6 = 14.42.4 * 6.9 = 14.4 + 2.4*0.9 = 14.4 + 2.16 = 16.56So, 16.64 - 16.56 = 0.08So, 0.08 / 2.4 ≈ 0.0333Thus, ȳ ≈ 6.9 + 0.0333 ≈ 6.9333So, the centroid is approximately at (5.7333, 6.9333)Now, according to Pappus's theorem, the volume is equal to the area A multiplied by the distance traveled by the centroid, which is 2π times the distance from the centroid to the axis of revolution.So, I need to find the distance from the centroid (5.7333, 6.9333) to the line 4y - x = 20.The formula for the distance from a point (x0, y0) to the line ax + by + c = 0 is |ax0 + by0 + c| / sqrt(a² + b²)First, let me write the line 4y - x = 20 in standard form: -x + 4y - 20 = 0So, a = -1, b = 4, c = -20So, distance d = | -1*5.7333 + 4*6.9333 - 20 | / sqrt( (-1)^2 + 4^2 ) = | -5.7333 + 27.7332 - 20 | / sqrt(1 + 16) = | (-5.7333 - 20) + 27.7332 | / sqrt(17)Wait, let me compute step by step:-1*5.7333 = -5.73334*6.9333 = 27.7332So, sum: -5.7333 + 27.7332 = 22.0Then subtract 20: 22.0 - 20 = 2.0So, absolute value is |2.0| = 2.0Denominator: sqrt(1 + 16) = sqrt(17)So, distance d = 2 / sqrt(17)Therefore, the distance traveled by the centroid is 2πd = 2π*(2 / sqrt(17)) = 4π / sqrt(17)But wait, no. Wait, the distance traveled is the circumference, which is 2π times the radius, which is the distance d. So, it's 2πd = 2π*(2 / sqrt(17)) = 4π / sqrt(17)But actually, wait, no. The distance from the centroid to the axis is d, so the distance traveled is 2πd.So, yes, it's 2π*(2 / sqrt(17)) = 4π / sqrt(17)But let me double-check:d = | -x + 4y - 20 | / sqrt(1 + 16) = | -5.7333 + 4*6.9333 - 20 | / sqrt(17)Compute numerator:-5.7333 + 27.7332 - 20 = (-5.7333 - 20) + 27.7332 = (-25.7333) + 27.7332 ≈ 2.0So, yes, numerator is 2.0, so d = 2 / sqrt(17)Thus, distance traveled is 2π*(2 / sqrt(17)) = 4π / sqrt(17)Therefore, volume V = A * (distance traveled) = 2.4 * (4π / sqrt(17)) = 9.6π / sqrt(17)But let me rationalize the denominator:9.6π / sqrt(17) = (9.6π * sqrt(17)) / 17But 9.6 is 96/10 = 48/5, so:V = (48/5 * π * sqrt(17)) / 17 = (48π sqrt(17)) / (5*17) = (48π sqrt(17)) / 85Simplify 48 and 85: they have no common factors, so that's the simplest form.So, V = (48π sqrt(17)) / 85Wait, but let me check my calculations again because I might have made a mistake in the centroid coordinates or the distance.Wait, earlier I approximated the centroid as (5.7333, 6.9333). Let me see if that's accurate.x̄ was 13.76 / 2.4 ≈ 5.7333ȳ was 16.64 / 2.4 ≈ 6.9333But let me compute these more precisely.x̄ = 13.76 / 2.413.76 ÷ 2.4:2.4 * 5 = 1213.76 - 12 = 1.761.76 / 2.4 = 0.7333...So, x̄ = 5.7333...Similarly, ȳ = 16.64 / 2.416.64 ÷ 2.4:2.4 * 6 = 14.416.64 - 14.4 = 2.242.24 / 2.4 = 0.9333...So, ȳ = 6.9333...So, the centroid is indeed at (5.7333, 6.9333)Now, computing the distance:d = | -5.7333 + 4*6.9333 - 20 | / sqrt(17)Compute numerator:-5.7333 + 27.7332 - 20 = (-5.7333 - 20) + 27.7332 = (-25.7333) + 27.7332 ≈ 2.0So, d = 2 / sqrt(17)Thus, distance traveled is 2π*(2 / sqrt(17)) = 4π / sqrt(17)So, volume V = A * (distance traveled) = 2.4 * (4π / sqrt(17)) = 9.6π / sqrt(17)Convert 9.6 to fraction: 9.6 = 96/10 = 48/5So, V = (48/5)π / sqrt(17) = (48π) / (5 sqrt(17)) = (48π sqrt(17)) / (5*17) = (48π sqrt(17)) / 85Simplify 48 and 85: GCD is 1, so it's already simplified.So, V = (48π sqrt(17)) / 85But let me check if I did everything correctly. Maybe I made a mistake in the centroid calculation.Wait, when I computed x̄, I had:x̄ = (1/A) * [ ∫ x dA ] = (1/2.4)*(13.76) ≈ 5.7333Similarly for ȳ.But let me verify the integrals again.First, for x̄:First integral: ∫ from 4 to 6 x*(3x/4 - 3) dx = ∫ (3x²/4 - 3x) dx from 4 to 6Antiderivative: (x³/4 - (3/2)x²)At 6: (216/4 - (3/2)*36) = 54 - 54 = 0At 4: (64/4 - (3/2)*16) = 16 - 24 = -8So, integral is 0 - (-8) = 8Second integral: ∫ from 6 to 7.2 x*(9 - 5x/4) dx = ∫ (9x - 5x²/4) dxAntiderivative: (9/2)x² - (5/12)x³At 7.2: (9/2)*(51.84) - (5/12)*(373.248) ≈ 233.28 - 155.52 = 77.76At 6: (9/2)*36 - (5/12)*216 = 162 - 90 = 72So, integral is 77.76 - 72 = 5.76Total x̄ integral: 8 + 5.76 = 13.76So, x̄ = 13.76 / 2.4 ≈ 5.7333Similarly for ȳ:First integral: ∫ from 4 to 6 (15x² + 24x - 336)/32 dxAntiderivative: (5x³ + 12x² - 336x)/32At 6: (5*216 + 12*36 - 336*6)/32 = (1080 + 432 - 2016)/32 = (-504)/32 = -15.75At 4: (5*64 + 12*16 - 336*4)/32 = (320 + 192 - 1344)/32 = (-832)/32 = -26So, integral is (-15.75) - (-26) = 10.25Second integral: ∫ from 6 to 7.2 (15x² - 488x + 2736)/32 dxAntiderivative: (5x³ - 244x² + 2736x)/32At 7.2: (5*373.248 - 244*51.84 + 2736*7.2)/32 ≈ (1866.24 - 12648.96 + 19699.2)/32 ≈ (8916.48)/32 ≈ 278.64At 6: (5*216 - 244*36 + 2736*6)/32 = (1080 - 8784 + 16416)/32 = (8712)/32 = 272.25So, integral is 278.64 - 272.25 ≈ 6.39Total ȳ integral: 10.25 + 6.39 ≈ 16.64ȳ = 16.64 / 2.4 ≈ 6.9333So, the centroid is indeed at approximately (5.7333, 6.9333)Distance from centroid to the line 4y - x = 20 is 2 / sqrt(17)Thus, volume V = 2.4 * (4π / sqrt(17)) = 9.6π / sqrt(17) = (48π sqrt(17)) / 85But let me check if I can simplify this further or if I made a mistake in the calculation.Wait, 48 and 85 have no common factors, so that's the simplest form.Alternatively, I can write it as (48/85)π sqrt(17)But perhaps the problem expects an exact form, so I think (48π sqrt(17))/85 is acceptable.Wait, but let me check if I did the distance correctly.The line is 4y - x = 20, which is equivalent to -x + 4y - 20 = 0So, a = -1, b = 4, c = -20Point is (5.7333, 6.9333)Compute | -1*5.7333 + 4*6.9333 - 20 | / sqrt(1 + 16)Compute numerator:-5.7333 + 27.7332 - 20 = (-5.7333 - 20) + 27.7332 = (-25.7333) + 27.7332 ≈ 2.0So, numerator is 2.0Denominator is sqrt(17)Thus, distance d = 2 / sqrt(17)So, distance traveled is 2πd = 2π*(2 / sqrt(17)) = 4π / sqrt(17)Thus, volume V = A * (distance traveled) = 2.4 * (4π / sqrt(17)) = 9.6π / sqrt(17)But 9.6 is 48/5, so:V = (48/5)π / sqrt(17) = (48π) / (5 sqrt(17)) = (48π sqrt(17)) / (5*17) = (48π sqrt(17)) / 85Yes, that's correct.But wait, I think I made a mistake in the area calculation. Earlier, I got A = 2.4, but let me double-check.Area A = ∫ from 4 to 6 (3x/4 - 3) dx + ∫ from 6 to 7.2 (9 - 5x/4) dxFirst integral:∫ from 4 to 6 (3x/4 - 3) dx = [ (3/8)x² - 3x ] from 4 to 6At 6: (3/8)*36 - 18 = 13.5 - 18 = -4.5At 4: (3/8)*16 - 12 = 6 - 12 = -6Difference: -4.5 - (-6) = 1.5Second integral:∫ from 6 to 7.2 (9 - 5x/4) dx = [9x - (5/8)x² ] from 6 to 7.2At 7.2: 9*7.2 = 64.8; (5/8)*(7.2)^2 = (5/8)*51.84 = 32.4; so 64.8 - 32.4 = 32.4At 6: 9*6 = 54; (5/8)*36 = 22.5; so 54 - 22.5 = 31.5Difference: 32.4 - 31.5 = 0.9Total area: 1.5 + 0.9 = 2.4Yes, that's correct.So, everything checks out. Therefore, the volume is (48π sqrt(17))/85But let me see if I can simplify this fraction:48 and 85: GCD is 1, so it's already in simplest terms.Alternatively, I can write it as (48/85)π sqrt(17)But perhaps the problem expects a rationalized form, so (48π sqrt(17))/85 is fine.Wait, but let me check if I can write it as (48π)/sqrt(17) * (sqrt(17)/85) = (48π sqrt(17))/85, which is the same.So, I think that's the final answer.**Final Answer**The volume of the solid is boxed{dfrac{48sqrt{17}}{85}pi}.

answer：Okay, so I have this proposition p: There exists an x₀ in the interval (0, +∞) such that (1/2) minus 2 to the power of negative x₀ equals 5/8. I need to find the negation of p. Hmm, let's see.First, I remember that in logic, the negation of an existential quantifier (∃) is a universal quantifier (∀). So, if the original statement is saying "there exists an x₀ such that something is true," then the negation would be "for all x, that something is not true."So, applying that here, the negation of p should be: For all x in (0, +∞), (1/2) minus 2^(-x) is not equal to 5/8.Wait, let me make sure I'm doing this right. The original statement is ∃x₀ ∈ (0, +∞), (1/2) - 2^(-x₀) = 5/8. So, the negation would indeed be ∀x ∈ (0, +∞), (1/2) - 2^(-x) ≠ 5/8.Is there anything else I need to consider? Maybe I should check if the equation actually has a solution to understand the context better. Let's solve (1/2) - 2^(-x) = 5/8.Subtracting 1/2 from both sides gives -2^(-x) = 5/8 - 1/2. 5/8 minus 1/2 is 5/8 - 4/8 = 1/8. So, -2^(-x) = 1/8. Multiplying both sides by -1, we get 2^(-x) = -1/8.But 2^(-x) is always positive because any positive number raised to any power is positive. So, 2^(-x) can never be negative, which means there's no solution to the equation. Therefore, the original proposition p is false because there's no such x₀ in (0, +∞) that satisfies the equation.But wait, the question is just asking for the negation of p, not whether p is true or false. So, regardless of whether p is true or false, the negation is simply the statement that for all x in (0, +∞), (1/2) - 2^(-x) is not equal to 5/8.I think that's it. I don't see any other steps needed here. The negation process seems straightforward once I recall how to negate existential quantifiers.

answer：Alright, let's tackle this problem step by step. So, we have the function ( f(x) = log_a |x + 1| ), and we're told that it's greater than 0 in the interval ((-1, 0)). We need to figure out which of the options A, B, C, or D is true regarding whether the function is increasing or decreasing on certain intervals.First, let's understand what the function looks like. The function is a logarithm with base ( a ), and the argument is ( |x + 1| ). Since the logarithm function is only defined for positive arguments, ( |x + 1| ) must be positive, which it is for all ( x ) except ( x = -1 ). So, our domain is all real numbers except ( x = -1 ).Now, the function ( f(x) ) is given to be greater than 0 in the interval ((-1, 0)). Let's think about what that means for the base ( a ). Remember, the logarithm function ( log_a y ) is greater than 0 when:- ( y > 1 ) and ( a > 1 ), or- ( 0 < y < 1 ) and ( 0 < a < 1 ).In our case, ( y = |x + 1| ). Let's analyze ( |x + 1| ) in the interval ((-1, 0)). If ( x ) is between (-1) and (0), then ( x + 1 ) is between (0) and (1). Therefore, ( |x + 1| ) is between (0) and (1). So, ( y ) is between (0) and (1).Given that ( f(x) = log_a y > 0 ) in this interval, and since ( y ) is between (0) and (1), this implies that the base ( a ) must be between (0) and (1). Because if ( a ) were greater than (1), ( log_a y ) would be negative when ( y ) is between (0) and (1). So, we can conclude that ( 0 < a < 1 ).Now, knowing that ( 0 < a < 1 ), let's recall the properties of logarithmic functions. For ( 0 < a < 1 ), the function ( log_a y ) is a decreasing function. That is, as ( y ) increases, ( log_a y ) decreases, and vice versa.Next, let's consider the inner function ( |x + 1| ). This is a V-shaped graph that is shifted one unit to the left. It has a vertex at ( x = -1 ). To the left of ( x = -1 ), the function ( |x + 1| ) is decreasing, and to the right of ( x = -1 ), it's increasing.So, breaking down the domain:1. For ( x < -1 ), ( |x + 1| = -(x + 1) = -x - 1 ), which is a linear function with a negative slope, meaning it's decreasing as ( x ) increases.2. For ( x > -1 ), ( |x + 1| = x + 1 ), which is a linear function with a positive slope, meaning it's increasing as ( x ) increases.Now, let's analyze the behavior of ( f(x) = log_a |x + 1| ) in these intervals.Starting with ( x < -1 ):- Here, ( |x + 1| ) is decreasing as ( x ) increases.- Since ( log_a y ) is a decreasing function (because ( 0 < a < 1 )), when ( y ) decreases, ( log_a y ) increases.- Therefore, as ( x ) increases from ( -infty ) to ( -1 ), ( |x + 1| ) decreases, so ( log_a |x + 1| ) increases.- This means ( f(x) ) is increasing on ( (-infty, -1) ).Now, for ( x > -1 ):- Here, ( |x + 1| ) is increasing as ( x ) increases.- Again, since ( log_a y ) is decreasing, when ( y ) increases, ( log_a y ) decreases.- Therefore, as ( x ) increases from ( -1 ) to ( infty ), ( |x + 1| ) increases, so ( log_a |x + 1| ) decreases.- This means ( f(x) ) is decreasing on ( (-1, infty) ).But wait, the problem specifically mentions the interval ((-1, 0)). We already considered that ( f(x) > 0 ) in this interval, which helped us determine that ( 0 < a < 1 ). Now, looking at the options given:A: ( f(x) ) is an increasing function on ( (-infty, 0) ).B: ( f(x) ) is a decreasing function on ( (-infty, 0) ).C: ( f(x) ) is an increasing function on ( (-infty, -1) ).D: ( f(x) ) is a decreasing function on ( (-infty, -0) ).Let's evaluate each option.Option A: ( f(x) ) is increasing on ( (-infty, 0) ).- From our analysis, ( f(x) ) is increasing on ( (-infty, -1) ) and decreasing on ( (-1, 0) ). So, overall on ( (-infty, 0) ), the function is not consistently increasing. It increases up to ( x = -1 ) and then decreases from ( x = -1 ) to ( x = 0 ). Therefore, Option A is incorrect.Option B: ( f(x) ) is decreasing on ( (-infty, 0) ).- Similar reasoning as above. Since ( f(x) ) increases on ( (-infty, -1) ) and decreases on ( (-1, 0) ), it's not consistently decreasing on the entire interval ( (-infty, 0) ). So, Option B is also incorrect.Option C: ( f(x) ) is increasing on ( (-infty, -1) ).- This aligns perfectly with our earlier conclusion. On ( (-infty, -1) ), ( f(x) ) is indeed increasing. Therefore, Option C is correct.Option D: ( f(x) ) is decreasing on ( (-infty, -0) ).- Wait, ( (-infty, -0) ) is the same as ( (-infty, 0) ). So, this is essentially the same as Option B, which we've already determined is incorrect. Therefore, Option D is also incorrect.So, after evaluating all options, only Option C is correct.**Final Answer**boxed{C}

answer：Okay, so I'm trying to figure out the probability that player A does not lose in a Chinese chess competition. The problem says that the probability of A winning is 0.5, and the probability of a draw is 0.2. Hmm, okay, so I need to find the probability that A either wins or draws, right? Because if A doesn't lose, that means A either wins or the game ends in a draw.Wait, let me think about this again. So, in a game between two players, there are three possible outcomes: A wins, B wins, or it's a draw. The problem gives me the probabilities for A winning and for a draw. It doesn't mention the probability of B winning directly, but maybe I don't need it for this particular question.So, the question is asking for the probability that A does not lose. That means A either wins or the game is a draw. So, I need to add the probability of A winning to the probability of a draw. That makes sense because these are two separate events that both result in A not losing.Let me write that down. The probability of A winning is 0.5, and the probability of a draw is 0.2. So, if I add those together, I should get the probability that A does not lose. So, 0.5 plus 0.2 equals 0.7. Therefore, the probability that A does not lose is 0.7.Wait a second, is there any chance that these two events overlap or something? Like, could A both win and draw at the same time? No, that doesn't make sense. In a single game, A can either win, lose, or draw. So, winning and drawing are mutually exclusive events. That means I can safely add their probabilities without worrying about double-counting any outcomes.Also, just to be thorough, let me check if the total probability adds up to 1. The probability of A winning is 0.5, the probability of a draw is 0.2, so the probability of B winning must be 1 minus 0.5 minus 0.2, which is 0.3. So, 0.5 plus 0.2 plus 0.3 equals 1, which makes sense because those are the only possible outcomes.Therefore, I'm confident that adding the probability of A winning and the probability of a draw gives me the correct probability that A does not lose. So, 0.5 plus 0.2 equals 0.7. Yeah, that seems right.I guess another way to think about it is that "does not lose" is the complement of "loses." So, if I can find the probability that A loses, which is the probability that B wins, and then subtract that from 1, I should get the same result. The probability of B winning is 0.3, so 1 minus 0.3 is 0.7. Yep, that matches what I got before.So, both methods give me the same answer, which is reassuring. I think I've got it right.

answer：Okay, so I have this problem: I need to find the number of positive integers ( n leq 1200 ) such that ( 12n ) is a perfect square. Hmm, let me try to figure this out step by step.First, I know that a perfect square is a number that can be expressed as some integer squared, like ( 1, 4, 9, 16, ) and so on. So, ( 12n ) needs to be one of these perfect squares. Let me denote that perfect square as ( k^2 ), where ( k ) is a positive integer. Therefore, I can write:[ 12n = k^2 ]From this equation, I can solve for ( n ):[ n = frac{k^2}{12} ]Since ( n ) has to be a positive integer, ( frac{k^2}{12} ) must also be an integer. That means ( k^2 ) must be divisible by 12. Let me think about the prime factors of 12. The prime factorization of 12 is:[ 12 = 2^2 times 3 ]So, ( k^2 ) must have at least the prime factors ( 2^2 ) and ( 3 ). But wait, in a perfect square, all the exponents in the prime factorization must be even. That means if ( k^2 ) has a factor of 3, it must have at least ( 3^2 ) to make the exponent even. Similarly, for 2, it already has ( 2^2 ), which is fine because 2 is already squared.So, putting that together, ( k^2 ) must have at least ( 2^2 times 3^2 ) as factors. Therefore, ( k ) itself must have at least ( 2 times 3 ) as factors, because when you square ( k ), the exponents double. So, ( k ) must be a multiple of 6. Let me write that as:[ k = 6m ]where ( m ) is a positive integer. Substituting this back into the equation for ( n ):[ n = frac{(6m)^2}{12} = frac{36m^2}{12} = 3m^2 ]So, ( n ) must be equal to ( 3m^2 ). Now, since ( n leq 1200 ), I can write:[ 3m^2 leq 1200 ]Dividing both sides by 3:[ m^2 leq 400 ]Taking the square root of both sides:[ m leq sqrt{400} ][ m leq 20 ]So, ( m ) can be any positive integer from 1 up to 20. That gives me 20 possible values for ( m ), right? Wait, hold on. Let me double-check.If ( m = 1 ), then ( n = 3(1)^2 = 3 ). If ( m = 20 ), then ( n = 3(20)^2 = 3 times 400 = 1200 ). So, yes, ( m ) can indeed range from 1 to 20, inclusive. That gives me 20 possible values for ( m ), and hence 20 possible values for ( n ).But wait, earlier I thought ( k ) had to be a multiple of 6, but when I substituted ( k = 6m ), I ended up with ( n = 3m^2 ). Is that correct? Let me verify with an example.Take ( m = 1 ): ( n = 3 times 1 = 3 ). Then, ( 12n = 12 times 3 = 36 ), which is ( 6^2 ), a perfect square. Good.Take ( m = 2 ): ( n = 3 times 4 = 12 ). Then, ( 12n = 12 times 12 = 144 ), which is ( 12^2 ), a perfect square. Perfect.Take ( m = 3 ): ( n = 3 times 9 = 27 ). Then, ( 12n = 12 times 27 = 324 ), which is ( 18^2 ), a perfect square. Nice.Wait a second, but in my initial substitution, I set ( k = 6m ), so ( k ) is 6, 12, 18, etc., which are multiples of 6. Then, ( k^2 ) is 36, 144, 324, etc., which are indeed perfect squares, and when divided by 12, give ( n ) as 3, 12, 27, etc. So, that seems consistent.But earlier, I thought that ( k ) must be a multiple of 6 because ( k^2 ) must have at least ( 2^2 times 3^2 ) as factors. So, that reasoning seems correct.But wait, hold on. Let me think again about the prime factors. If ( 12n = k^2 ), then the prime factors of ( 12n ) must all have even exponents. The prime factors of 12 are ( 2^2 times 3 ). So, when multiplied by ( n ), which has its own prime factors, the result must have even exponents for all primes.Therefore, ( n ) must supply the necessary factors to make all exponents even. Specifically, since 12 already has ( 2^2 ) (which is even) and ( 3^1 ) (which is odd), ( n ) must have at least one more factor of 3 to make it ( 3^2 ), and any other prime factors must come in pairs to maintain even exponents.So, ( n ) must be of the form ( 3 times text{(a perfect square)} ). That is, ( n = 3m^2 ), where ( m ) is a positive integer. That aligns with what I found earlier.Therefore, substituting back, ( n = 3m^2 leq 1200 ), so ( m^2 leq 400 ), hence ( m leq 20 ). So, ( m ) can be 1, 2, 3, ..., 20, giving 20 possible values for ( m ), and thus 20 possible values for ( n ).Wait, but in the initial problem, the user wrote:"Find the number of positive integers ( n leq 1200 ) such that ( 12n ) is a perfect square."And in the thought process, I concluded 20. But in the initial answer, the assistant concluded 10. Hmm, that's conflicting.Let me check the assistant's reasoning:"Since ( 12 ) divides ( 12n ), and ( 12 ) is square-free (divisors 2 and 3 are prime), we must have ( 12^2 mid 12n ), so ( 12 mid n ). Let ( n = 12a ). Then ( 12^2 a = 12n ) is a square, and conversely, if ( a ) is a square, then ( 12^2 a ) is a square. Thus, we need to count the number of positive squares ( a ) such that ( 12a leq 1200 ) or ( a leq 100 ). The largest such square is ( 100 = 10^2 ), so the possible values of ( a ) are ( b^2 ) for ( b = 1, 2, 3, ..., 10 ), giving ( 10 ) possible values for ( a ) (and hence 10 possible values for ( n )."Wait, so the assistant is saying that ( n = 12a ), and ( a ) must be a perfect square, leading to ( a leq 100 ), so ( a ) can be ( 1^2, 2^2, ..., 10^2 ), hence 10 values.But in my reasoning, I concluded that ( n = 3m^2 ), leading to ( m leq 20 ), hence 20 values.So, which one is correct? Let me see.Let me test with ( n = 3m^2 ). For ( m = 1 ), ( n = 3 ), ( 12n = 36 = 6^2 ). Good.For ( m = 2 ), ( n = 12 ), ( 12n = 144 = 12^2 ). Good.For ( m = 3 ), ( n = 27 ), ( 12n = 324 = 18^2 ). Good.For ( m = 4 ), ( n = 48 ), ( 12n = 576 = 24^2 ). Good.For ( m = 5 ), ( n = 75 ), ( 12n = 900 = 30^2 ). Good.For ( m = 6 ), ( n = 108 ), ( 12n = 1296 = 36^2 ). Good.For ( m = 7 ), ( n = 147 ), ( 12n = 1764 = 42^2 ). Good.For ( m = 8 ), ( n = 192 ), ( 12n = 2304 = 48^2 ). Good.For ( m = 9 ), ( n = 243 ), ( 12n = 2916 = 54^2 ). Good.For ( m = 10 ), ( n = 300 ), ( 12n = 3600 = 60^2 ). Good.Wait, but ( m = 10 ) gives ( n = 300 ), which is less than 1200. So, why does the assistant say only 10 values? Because in the assistant's reasoning, ( a ) is a square, and ( a leq 100 ), so ( a ) can be up to ( 10^2 ), hence 10 values.But in my reasoning, ( m ) goes up to 20, giving 20 values. So, which is correct?Wait, perhaps the assistant made a mistake. Let me see.In the assistant's reasoning, they set ( n = 12a ), so ( 12n = 12 times 12a = 144a ). Then, they say that ( 144a ) must be a perfect square. Since 144 is already a perfect square (( 12^2 )), then ( a ) must be a perfect square as well. Therefore, ( a = b^2 ), and ( 12a leq 1200 ) implies ( a leq 100 ), so ( b^2 leq 100 ), hence ( b leq 10 ), giving 10 values.But in my reasoning, I set ( n = 3m^2 ), leading to ( m leq 20 ), hence 20 values.Wait, so which is correct? Let me check with ( m = 11 ): ( n = 3 times 121 = 363 ), ( 12n = 4356 ). Is 4356 a perfect square? Let me check: ( 66^2 = 4356 ). Yes, it is. So, ( n = 363 ) is valid, but according to the assistant's reasoning, ( a = 363 / 12 = 30.25 ), which is not an integer. So, that's a problem.Wait, so in the assistant's reasoning, ( n = 12a ), so ( a = n / 12 ). If ( n = 363 ), then ( a = 363 / 12 = 30.25 ), which is not an integer. Therefore, ( a ) must be an integer, so ( n ) must be a multiple of 12. But in my reasoning, ( n = 3m^2 ), which doesn't necessarily have to be a multiple of 12.Wait, so perhaps the assistant's reasoning is flawed because they assumed ( n = 12a ), but in reality, ( n ) doesn't have to be a multiple of 12. It just has to make ( 12n ) a perfect square. So, ( n ) can be 3, 12, 27, 48, etc., which are not all multiples of 12. For example, 3 is not a multiple of 12, but ( 12 times 3 = 36 ) is a perfect square.Therefore, the assistant's mistake was assuming that ( n ) must be a multiple of 12, which is not necessarily true. Instead, ( n ) must be such that when multiplied by 12, the result is a perfect square. This can happen even if ( n ) is not a multiple of 12, as long as the product has all even exponents in its prime factorization.So, going back, my initial reasoning seems correct: ( n = 3m^2 ), leading to ( m leq 20 ), hence 20 possible values for ( n ).But let me double-check with another example. Take ( m = 11 ): ( n = 3 times 121 = 363 ). Then, ( 12n = 4356 ), which is ( 66^2 ). Correct. So, 363 is a valid ( n ), and it's not a multiple of 12, but it still satisfies the condition.Therefore, the assistant's answer of 10 is incorrect because they restricted ( n ) to be multiples of 12, which is not necessary. The correct number should be 20.Wait, but let me think again. Maybe I'm missing something. Let me re-examine the prime factorization approach.Given ( 12n = k^2 ), and ( 12 = 2^2 times 3 ). So, ( 12n = 2^2 times 3 times n ). For this to be a perfect square, all primes in the factorization must have even exponents.So, in ( 2^2 times 3 times n ), the exponent of 2 is already 2 (even), but the exponent of 3 is 1 (odd). Therefore, ( n ) must supply another factor of 3 to make it ( 3^2 ), and any other prime factors in ( n ) must come in pairs to ensure even exponents.Therefore, ( n ) must be of the form ( 3 times m^2 ), where ( m ) is a positive integer. So, ( n = 3m^2 ). Then, ( 12n = 12 times 3m^2 = 36m^2 = (6m)^2 ), which is indeed a perfect square.Therefore, ( n = 3m^2 leq 1200 ) implies ( m^2 leq 400 ), so ( m leq 20 ). Hence, ( m ) can be 1, 2, 3, ..., 20, giving 20 possible values for ( n ).So, the correct answer should be 20, not 10. The assistant's mistake was in assuming ( n ) had to be a multiple of 12, which led them to an incorrect count.But wait, let me check the assistant's reasoning again:"Since ( 12 ) divides ( 12n ), and ( 12 ) is square-free (divisors 2 and 3 are prime), we must have ( 12^2 mid 12n ), so ( 12 mid n ). Let ( n = 12a ). Then ( 12^2 a = 12n ) is a square, and conversely, if ( a ) is a square, then ( 12^2 a ) is a square. Thus, we need to count the number of positive squares ( a ) such that ( 12a leq 1200 ) or ( a leq 100 ). The largest such square is ( 100 = 10^2 ), so the possible values of ( a ) are ( b^2 ) for ( b = 1, 2, 3, ..., 10 ), giving ( 10 ) possible values for ( a ) (and hence 10 possible values for ( n )."So, the assistant's reasoning is that since ( 12 ) is square-free, ( 12^2 ) must divide ( 12n ), implying ( 12 ) divides ( n ). But is that necessarily true?Wait, let's think about it. If ( 12n ) is a perfect square, and ( 12 ) is square-free, does that imply ( 12^2 ) divides ( 12n )?Well, ( 12n ) being a perfect square means that all prime exponents in its factorization are even. Since ( 12 = 2^2 times 3 ), the exponent of 2 is already 2 (even), but the exponent of 3 is 1 (odd). Therefore, to make ( 12n ) a perfect square, ( n ) must contribute another factor of 3, making the exponent of 3 equal to 2. However, ( n ) doesn't necessarily have to contribute another factor of 2, because the exponent of 2 is already even.Therefore, ( n ) must be divisible by 3, but not necessarily by 2. So, ( n ) must be a multiple of 3, but not necessarily a multiple of 12. Hence, the assistant's assumption that ( 12 ) divides ( n ) is incorrect. Instead, ( n ) only needs to be divisible by 3.Therefore, the correct approach is to let ( n = 3m^2 ), leading to ( m leq 20 ), hence 20 possible values for ( n ).So, in conclusion, the correct number of positive integers ( n leq 1200 ) such that ( 12n ) is a perfect square is 20, not 10.

answer：First, I need to determine the smallest integer that is greater than the square root of 10. Since (sqrt{10}) is approximately 3.16, the smallest integer greater than this value is 4.Next, I need to find the largest integer that is less than the square root of 75. Since (sqrt{75}) is approximately 8.66, the largest integer less than this value is 8.Now, I will list the integers between 4 and 8, inclusive: 4, 5, 6, 7, 8.Finally, I will count these integers to determine how many there are. There are 5 integers in total between (sqrt{10}) and (sqrt{75}) on the number line.