Question

Use the method of cylindrical shells to find the volume v generated by rotating the region bounded by the given curves about the specified axis. X = 5y2, y ≥ 0, x = 5; about y = 2.

Answer 1

The volume of the solid of revolution is $\frac{7\pi}{3}$ cubic units.

How to find the volume of a solid of revolution with respect to an axis parallel to a Cartesian axis

The statement has been represented in the image attached below, the formula for the solid of revolution is presented below:

$V = \pi \int\limits^{1}_{0} {[5\cdot y^{2}-2]^{2}} \, dy$ (1)

$V = \pi \int\limits^{1}_{0} {(25\cdot y^{4}-20\cdot y +4)} \, dy$

$V = 25\pi\int\limits^1_0 {y^{4}} \, dy -20\pi\int\limits^1_0 {y^{2}} \, dy +4\pi\int\limits^1_0\, dy$

$V = \frac{7\pi}{3}$

The volume of the solid of revolution is $\frac{7\pi}{3}$ cubic units. $\blacksquare$

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