Let $R$ be the leftmost region in the first quadrant enclosed by the $x$-axis and the curve $y=\sin (x)$.

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

Question

Let $R$ be the leftmost region in the first quadrant enclosed by the $x$-axis and the curve $y=\sin (x)$.

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

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ To find the volume of the solid generated by rotating the region $R$ about the line $y=1$, we use the washer method. The volume $V$ is given by the integral $V = \pi \int_{a}^{b} \left[ R_{ ext{outer}}^2 - R_{ ext{inner}}^2 ight] dx$. ‖ ‖ step 2 ⋮ Here, $R_{ ext{outer}} = 1$ (the distance from $y=1$ to the x-axis) and $R_{ ext{inner}} = 1 - \sin(x)$ (the distance from $y=1$ to the curve $y=\sin(x)$). ‖ ‖ step 3 ⋮ Therefore, the volume integral becomes $V = \pi \int_{0}^{\pi} \left[ 1^2 - (1 - \sin(x))^2 ight] dx$. ‖ ‖ step 4 ⋮ Simplifying the integrand: $1^2 - (1 - \sin(x))^2 = 1 - (1 - 2\sin(x) + \sin^2(x)) = 2\sin(x) - \sin^2(x)$. ‖ ‖ step 5 ⋮ Thus, the integral for the volume is $V = \pi \int_{0}^{\pi} \left[ 2\sin(x) - \sin^2(x) ight] dx$. ‖ ***A*** ∻Key Concept∻ ⚹Washer Method⚹ ∻Explanation∻ ⚹The washer method is used to find the volume of a solid of revolution when the region is rotated about a line that is not the x-axis or y-axis. It involves subtracting the volume of the inner solid from the volume of the outer solid.⚹◊What is the formula for finding the volume of a solid generated by rotating a region about the x-axis?⍭ Generate me a similar question◊

Sia