Compute the volume of the solid obtained by rotating the region bounded by $y = x^2$, $x = 0$, and $y = 4$ about the y-axis.

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∻Solution by Steps∻ ‖ step 1 ⋮ To find the volume of the solid obtained by rotating the region bounded by $y = x^2$, $x = 0$, and $y = 4$ about the y-axis, we use the method of disks or washers. ‖ ‖ step 2 ⋮ The volume $V$ of the solid of revolution about the y-axis is given by the integral $V = \int_{a}^{b} \pi r^2 dy$, where $r$ is the radius of a disk at a given $y$ and $[a, b]$ is the interval for $y$. ‖ ‖ step 3 ⋮ Since $y = x^2$, we solve for $x$ to find the radius: $x = \sqrt{y}$. Thus, the radius of the disk is $r = \sqrt{y}$. ‖ ‖ step 4 ⋮ The limits of integration are from $y = 0$ to $y = 4$ because the region is bounded by $y = 4$. ‖ ‖ step 5 ⋮ The volume integral becomes $V = \int_{0}^{4} \pi (\sqrt{y})^2 dy = \int_{0}^{4} \pi y dy$. ‖ ‖ step 6 ⋮ Integrating $\pi y$ with respect to $y$ from 0 to 4, we get $V = \left[\frac{\pi y^2}{2}\right]_0^4$. ‖ ‖ step 7 ⋮ Evaluating the integral, we find $V = \frac{\pi 4^2}{2} - \frac{\pi 0^2}{2} = \frac{16\pi}{2} = 8\pi$. ‖ ∻Answer∻ ⚹ The volume of the solid is $8\pi$ cubic units. ⚹ ∻Key Concept∻ ⚹Volume of a solid of revolution using the disk method⚹ ∻Explanation∻ ⚹The volume is found by integrating the area of circular disks, which are formed by rotating the region about the y-axis. The radius of each disk is given by the function describing the boundary of the region, in this case, $y = x^2$.⚹◊What is the formula for finding the volume of a solid obtained by rotating a region bounded by a curve around the y-axis?⍭ Generate me a similar question◊

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