AP Calculus AB Exam Question:

A curve is defined by the equation $y = x^2$ for $0 \leq x \leq 2$. The region enclosed by this curve is rotated about the x-axis to create a solid.

(a) Determine the volume of the solid of revolution generated by rotating the region about the x-axis.

(b) Determine the volume of the solid formed when the region is rotated about the line $y = -1$.

Step-by-Step Solution:

(a) Volume of the Solid of Revolution around the x-axis:

To find the volume of the solid of revolution, we use the disk method. The volume of each disk is given by the area of the circular disk, which is π times the radius squared.

In this case, the radius of each circular disk is given by the height of the curve, which is $y = x^2$. Thus, the radius is $r = x^2$.

We need to integrate the sum of all the infinitesimally thin disks from $x = 0$ to $x = 2$:

Substitute $r(x) = x^2$ into the integral:

Integrate:

Evaluate at the limits:

Simplify:

The volume of the solid of revolution around the x-axis is approximately $\frac{32 \pi}{5}$ units cubed.

(b) Volume of the Solid of Revolution around the line y = -1:

To find the volume of the solid of revolution around the line $y = -1$, we must adjust the radius of each circular disk. The radius of each disk is the perpendicular distance between the curve $y = x^2$ and the line $y = -1$.

Subtracting the equation of the line from the curve, we get $x^2 - (-1) = x^2 + 1$.

So, the radius is $r = x^2 + 1$.

We need to integrate the sum of all the infinitesimally thin disks from $x = 0$ to $x = 2$:

Substitute $r(x) = x^2 + 1$ into the integral:

Expand and simplify:

Integrate:

Evaluate at the limits:

Simplify:

The volume of the solid of revolution around the line $y = -1$ is approximately $\frac{1424 \pi}{75}$ units cubed.