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 cylindrical shells. ‖ ‖ step 2 ⋮ The volume of a thin cylindrical shell with radius $r$, height $h$, and thickness $\Delta r$ is given by $2\pi rh \Delta r$. ‖ ‖ step 3 ⋮ For the given region, the radius of the shell is $r = x$, the height is $h = x^2$, and the thickness is $\Delta r = dx$. ‖ ‖ step 4 ⋮ The volume of the entire solid is the integral of the volume of the shells from $x = 0$ to $x = 2$, since $y = x^2$ and $y = 4$ intersect at $x = 2$. ‖ ‖ step 5 ⋮ The integral for the volume is $\int_0^2 2\pi x(x^2) dx = 2\pi \int_0^2 x^3 dx$. ‖ ‖ step 6 ⋮ Evaluating the integral, we get $2\pi \left[\frac{x^4}{4}\right]_0^2 = 2\pi \left[\frac{2^4}{4} - \frac{0^4}{4}\right] = 2\pi \left[\frac{16}{4}\right] = 2\pi [4] = 8\pi$. ‖ ∻Answer∻ ⚹ The volume of the solid is $8\pi$ cubic units. ⚹ ∻Key Concept∻ ⚹Volume by Cylindrical Shells⚹ ∻Explanation∻ ⚹The volume of a solid of revolution can be found using the method of cylindrical shells, where the volume is the integral of the circumferences of the shells times their respective heights and thicknesses.⚹◊What is the formula for finding the volume of a solid obtained by rotating a region around the y-axis?⍭ Generate me a similar question◊

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