Question

Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line.


`x = 2 √(3y)`

`x = 0`

`y = 7`
; about the y-axis

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Answer 1

on this one, we proceed the same as the other, we start off by graphing them, Check the picture below. Now, we have a solid horizontal line at y = 7, that'd be our boundary, and a parabola that has a vertex at the origin, that's our other boundary, from 0 to 7.

In this case, is a solid area from the curve to the axis of rotation, so we can simply use the "disk" method to get the volume, lemme do it in the same way as "area under the curve", assuming h(y) = 0 for the axis of rotation

`\textit{area under the curve}\\ \stackrel{f(y)}{2√(3y)}~~ - ~~\stackrel{h(y)}{0}\qquad \implies 2√(3y) \\\\\\ \displaystyle\int_(0)^7\pi \left( 2√(3y) \right)^2 dy\implies \int_(0)^7\pi(12y)dy\implies 12\pi \int_(0)^7 y\cdot dy \\\\\\ 12\pi \cdot \left. \cfrac{y^2}{2} \right]_(0)^7\implies12\pi \cdot \cfrac{49}{2}\implies 294\pi`