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 each cylindrical shell is given by $y = x^2$. The radius of each shell is $x$. ‖ ‖ step 3 ⋮ The volume of a thin shell is given by $2\pi \cdot ext{radius} \cdot ext{height} \cdot ext{thickness}$. Here, the thickness is $dx$. ‖ ‖ step 4 ⋮ The volume of the solid is obtained by integrating the volume of the thin shells from $x = 0$ to $x = 2$. ‖ ‖ step 5 ⋮ Set up the integral: $\int_{0}^{2} 2\pi x (x^2) \, dx = 2\pi \int_{0}^{2} x^3 \, dx$. ‖ ‖ step 6 ⋮ Evaluate the integral: $2\pi \int_{0}^{2} x^3 \, dx = 2\pi \left[ \frac{x^4}{4} ight]_{0}^{2} = 2\pi \left( \frac{2^4}{4} - \frac{0^4}{4} ight) = 2\pi \left( \frac{16}{4} ight) = 2\pi \cdot 4 = 8\pi$. ‖ ∻Answer∻ ⚹ The volume of the solid is $8\pi$ cubic units. ⚹ ∻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 thin cylindrical shells.◊What is the formula for finding the volume of a solid of revolution obtained by rotating a region bounded by a curve about the y-axis⍭ Generate me a similar question◊⚹

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