A solid object is formed by revolving the region bounded by the curve $y = x^2$ and the y-axis about the y-axis. Find the volume of the solid.

Solution:

To find the volume of the solid, we will use the method of cylindrical shells.

First, we need to determine the limits of integration. The curve $y = x^2$ intersects the y-axis at the point (0,0) and extends infinitely in the positive y-direction. Therefore, our limits of integration will be from 0 to infinity.

Next, let's consider an infinitesimally thin strip of width $dy$ at a distance $y$ from the y-axis. The length of this strip is given by $2\pi y$ (since it is formed by rotating a line segment of length $y$ around the y-axis).

The height of the strip can be obtained by finding the corresponding x-values on the curve $y = x^2$. Since we are revolving around the y-axis, these x-values will be given by $x = \sqrt{y}$. Therefore, the height of the strip is $\sqrt{y}$.

Hence, the volume of the solid can be approximated by summing up the volumes of all these cylindrical shells:

Integrating this expression, we have:

Now, let's evaluate the integral:

Plugging in the limits of integration, we get:

Since $(0)^{5/2} = 0$, the second term in the parentheses becomes zero.

Thus, the volume of the solid is:

Taking the limit as $a$ approaches infinity, we have:

Therefore, the volume of the solid is infinite.