Volumes of Solids of Revolution Method of Cylinders — Question 2

Question 2 – Solution

1. Choose horizontal slices and $y$-bounds.

Shell slices run parallel to the $x$-axis. Rewrite $y=\sqrt{x}$ as $x=y^2$. The region extends from $y=0$ to $y=\sqrt4=2$.

2. Identify the shell radius and height.

The radius is the distance to $y=0$. For a horizontal slice, shell height is its horizontal length, right minus left:$$r(y)=y,\qquad h(y)=4-y^2.$$

3. Set up the shell integral.

A thin shell has volume approximately $(2\pi r)(h)\,dy$. Thus$$V=2\pi\int_0^2 y(4-y^2)\,dy=2\pi\int_0^2(4y-y^3)\,dy.$$

4. Integrate with respect to $y$.

$$\int4y\,dy=2y^2,\qquad\int y^3\,dy=\frac{y^4}{4}.$$$$V=2\pi\left[2y^2-\frac{y^4}{4}\right]_0^2.$$

5. Evaluate both endpoints.

$$V=2\pi\left[\left(2(2)^2-\frac{2^4}{4}\right)-0\right]=2\pi(8-4)=\boxed{8\pi}.$$

All volumes are in cubic units.