Activity 6.2.4

In each of the following questions, draw a careful, labeled sketch of the region described, as well as the resulting solid that results from revolving the region about the stated axis. In addition, draw a representative slice and state the volume of that slice, along with a definite integral whose value is the volume of the entire solid. It is not necessary to evaluate the integrals you find. For each prompt, use the finite region $S$ in the first quadrant bounded by the curves $y = 2x$ and $y = x^3$.

1. Revolve $S$ about the line $y = -2$.

2. Revolve $S$ about the line $y = 4$.

3. Revolve $S$ about the line $x=-1$.

4. Revolve $S$ about the line $x = 5$.