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Use the integration capabilities of the graphing utility to approximate the volume of the solid generated by revolving the region about the y-axis.

y=(x2)2(x6)23,y=0,x=2,x=6y=\sqrt[3]{(x-2)^2(x-6)^2}, \quad y=0, \quad x=2, \quad x=6


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V=2πabp(x)h(x)dxV=2\pi \int_a^b p\left(x\right)h\left(x\right)dx

V=2πabp(x)h(x)dxApply Shell method.=2π26x(x2)2(x6)23dxSubstitute x for p(x) and(x2)2(x6)23 for h(x).2π29.82Use the IntegralCalculator.187.25\begin{aligned} V&=2\pi \int_a^b p\left(x\right)h\left(x\right)dx && \text{Apply Shell method.} \\ &=2\pi \int_2^6 x\sqrt[3]{\left(x-2\right)^2\left(x-6\right)^2} dx && \text{Substitute} \ x \ \text{for} \ p\left(x\right) \ \text{and} \\ &\text{} && \sqrt[3]{\left(x-2\right)^2\left(x-6\right)^2} \ \text{for} \ h\left(x\right). \\ &\approx 2\pi \cdot 29.82 && \text{Use the IntegralCalculator.} \\ &\approx 187.25 \end{aligned}

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