Find the volume of the solid of revolution obtained by rotating the curve $y = sin(x)$, from $x = 0$ to $x = pi$, about the x-axis.

Question

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ To find the volume of the solid of revolution obtained by rotating the curve $y = \sin(x)$ from $x = 0$ to $x = \pi$ about the x-axis, we use the disk method. The volume $V$ is given by the integral formula: $V = \pi \int_{0}^{\pi} [\sin(x)]^2 \, dx$. ‖ ‖ step 2 ⋮ We need to evaluate the integral $\int_{0}^{\pi} \sin^2(x) \, dx$. Using the trigonometric identity $\sin^2(x) = \frac{1 - \cos(2x)}{2}$, we rewrite the integral: $\int_{0}^{\pi} \sin^2(x) \, dx = \int_{0}^{\pi} \frac{1 - \cos(2x)}{2} \, dx$. ‖ ‖ step 3 ⋮ Split the integral into two parts: $\int_{0}^{\pi} \frac{1}{2} \, dx - \int_{0}^{\pi} \frac{\cos(2x)}{2} \, dx$. ‖ ‖ step 4 ⋮ Evaluate the first part: $\int_{0}^{\pi} \frac{1}{2} \, dx = \frac{1}{2} \int_{0}^{\pi} 1 \, dx = \frac{1}{2} [x]_{0}^{\pi} = \frac{\pi}{2}$. ‖ ‖ step 5 ⋮ Evaluate the second part: $\int_{0}^{\pi} \frac{\cos(2x)}{2} \, dx = \frac{1}{2} \int_{0}^{\pi} \cos(2x) \, dx$. ‖ ‖ step 6 ⋮ Use the substitution $u = 2x$, $du = 2dx$, so $dx = \frac{du}{2}$. The limits change from $x = 0$ to $x = \pi$ to $u = 0$ to $u = 2\pi$. The integral becomes $\frac{1}{2} \int_{0}^{2\pi} \cos(u) \frac{du}{2} = \frac{1}{4} \int_{0}^{2\pi} \cos(u) \, du$. ‖ ‖ step 7 ⋮ Evaluate the integral: $\int_{0}^{2\pi} \cos(u) \, du = [\sin(u)]_{0}^{2\pi} = \sin(2\pi) - \sin(0) = 0 - 0 = 0$. ‖ ‖ step 8 ⋮ Combine the results: $\int_{0}^{\pi} \sin^2(x) \, dx = \frac{\pi}{2} - 0 = \frac{\pi}{2}$. ‖ ‖ step 9 ⋮ Multiply by $\pi$ to find the volume: $V = \pi \int_{0}^{\pi} \sin^2(x) \, dx = \pi \cdot \frac{\pi}{2} = \frac{\pi^2}{2}$. ‖ ∻Answer∻ ⚹ The volume of the solid of revolution is $\frac{\pi^2}{2}$. ⚹ ∻Key Concept∻ ⚹Disk Method⚹ ∻Explanation∻ ⚹The disk method involves integrating the square of the function being rotated around the axis, multiplied by $\pi$, over the given interval.⚹◊What is the formula for finding the volume of a solid of revolution obtained by rotating a curve about the x-axis?⍭ Generate me a similar question◊

Sia