Let $R$ be the region enclosed by the line $y=2$ and the curve $2(x-1)^{2}$.

A solid is generated by rotating $R$ about the line $y=2$.
Which one of the definite integrals gives the volume of the solid?
Choose 1 answer:
(A) $\pi \int_{0}^{2}\left[2-2(x-1)^{2}\right] d x$
(B) $\pi \int_{0}^{2}\left[2(x-1)^{2}\right]^{2} d x$
(C) $\pi \int_{0}^{2}\left[2-2(x-1)^{2}\right]^{2} d x$
() $\pi \int_{0}^{2} 2(x-1)^{2} d x$

Question

Let $R$ be the region enclosed by the line $y=2$ and the curve $2(x-1)^{2}$.

A solid is generated by rotating $R$ about the line $y=2$.
Which one of the definite integrals gives the volume of the solid?
Choose 1 answer:
(A) $\pi \int_{0}^{2}\left[2-2(x-1)^{2}\right] d x$
(B) $\pi \int_{0}^{2}\left[2(x-1)^{2}\right]^{2} d x$
(C) $\pi \int_{0}^{2}\left[2-2(x-1)^{2}\right]^{2} d x$
() $\pi \int_{0}^{2} 2(x-1)^{2} d x$

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ We need to find the volume of the solid generated by rotating the region $R$ about the line $y=2$. ‖ ‖ step 2 ⋮ The region $R$ is enclosed by the line $y=2$ and the curve $y=2(x-1)^2$. ‖ ‖ step 3 ⋮ The volume of the solid of revolution can be found using the disk method. The formula for the volume is $V = \pi \int_{a}^{b} [R(x)]^2 dx$, where $R(x)$ is the distance from the axis of rotation to the curve. ‖ ‖ step 4 ⋮ Here, $R(x) = 2 - 2(x-1)^2$. ‖ ‖ step 5 ⋮ Therefore, the volume is given by $V = \pi \int_{0}^{2} [2 - 2(x-1)^2]^2 dx$. ‖ ‖ step 6 ⋮ This matches option (C) in the multiple choice answers. ‖ ***C*** ∻Key Concept∻ ⚹Volume of a solid of revolution⚹ ∻Explanation∻ ⚹The volume of a solid generated by rotating a region about a line can be found using the disk method, which involves integrating the square of the distance from the axis of rotation to the curve.⚹◊What is the formula for finding the volume of a solid generated by rotating a region enclosed by two functions about a given line⍭ Generate me a similar question◊

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