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 height of a typical shell is given by $h = y = x^2$. ‖ ‖ step 3 ⋮ The radius of a typical shell is $r = x$. ‖ ‖ step 4 ⋮ The volume of a typical shell is $dV = 2 \pi x \cdot h \cdot dx = 2 \pi x \cdot x^2 \cdot dx = 2 \pi x^3 \cdot dx$. ‖ ‖ step 5 ⋮ To find the total volume, we integrate $dV$ from $x = 0$ to $x = 2$ (since $y = 4$ corresponds to $x = 2$): ‖ ‖ step 6 ⋮ $V = \int_{0}^{2} 2 \pi x^3 \, dx$. ‖ ‖ step 7 ⋮ Compute the integral: ‖ ‖ step 8 ⋮ $V = 2 \pi \int_{0}^{2} x^3 \, dx = 2 \pi \left[ \frac{x^4}{4} \right]_{0}^{2} = 2 \pi \left( \frac{2^4}{4} - \frac{0^4}{4} \right) = 2 \pi \left( \frac{16}{4} \right) = 2 \pi \cdot 4 = 8 \pi$. ‖ ∻Answer∻ ⚹ The volume of the solid is $8 \pi$. ⚹ ∻Key Concept∻ ⚹Cylindrical Shells Method⚹ ∻Explanation∻ ⚹The cylindrical shells method is used to find the volume of a solid of revolution by integrating the volume of infinitesimally thin cylindrical shells.⚹◊What is the formula for finding the volume of a solid obtained by rotating a region bounded by a curve about the y-axis?⍭ Generate me a similar question◊

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