AP Calculus AB Exam Question

A solid is formed by rotating the region bounded by the curve $y = \sqrt{x}$ and the x-axis from $x = 0$ to $x = 4$ about the x-axis. Find the volume of the resulting solid.

Solution:

To find the volume of the solid of revolution, we can use the disk method. We can divide the region into infinitely many disks with thickness $\Delta x$ and radius equal to the corresponding value of $y$.

The formula for the volume of a single disk is given by:

In this case, the radius is equal to $y = \sqrt{x}$ and $\Delta x$ represents an infinitesimally small width.

To find the volume of the entire solid, we need to sum up the volumes of all the disks. We can use integration to do this.

The volume of the solid can be computed as follows:

In this case, the lower limit is $a = 0$ and the upper limit is $b = 4$. So, we have:

Simplifying,

Integrating $x$ with respect to $x$, we get:

Plugging in the limits of integration,

Simplifying,

Finally,

Therefore, the volume of the resulting solid is $8\pi$.