Mathematics

Mathematics

Problem

The region bounded by y=xy=x and y=2−xy=2-x for 0≤x≤20\le x\le2 is revolved about the line y=−1y=-1. Which of the following definite integrals equals the volume of the resulting solid? a) π∫02[(3−x)2−(x+1)2] dx\displaystyle \pi\int_{0}^{2}\bigl[(3-x)^{2}-(x+1)^{2}\bigr]\,dx b) 2π∫02[(3−x)−(x+1)] dx\displaystyle 2\pi\int_{0}^{2}\bigl[(3-x)-(x+1)\bigr]\,dx c) π[∫01[(3−x)2−(x+1)2] dx+∫12[(x+1)2−(3−x)2] dx]\displaystyle \pi\left[\int_{0}^{1}\bigl[(3-x)^{2}-(x+1)^{2}\bigr]\,dx+\int_{1}^{2}\bigl[(x+1)^{2}-(3-x)^{2}\bigr]\,dx\right] d) π[∫01(3−x)2 dx+∫12(x+1)2 dx]\displaystyle \pi\left[\int_{0}^{1}(3-x)^{2}\,dx+\int_{1}^{2}(x+1)^{2}\,dx\right]

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