To find the volume of the solid generated by rotating the region bounded by $y = x^2$ and $y = 4$ in the first quadrant about the $x$-axis, we can apply the disk method. Here's how we set up the integral step-by-step:

Step 1: Determine the limits of integration

The region in the first quadrant is bounded by $y = x^2$ and $y = 4$. We need to find the $x$-values where these two curves intersect.

Set $x^2 = 4$: $x = \pm 2$ Since we're only considering the first quadrant, the intersection occurs at $x = 2$.

Thus, the limits of integration will be from $x = 0$ to $x = 2$.

Step 2: Set up the formula using the disk method

The formula for the volume of a solid of revolution using the disk method is: $V = \pi \int_{a}^{b} [R(x)]^2 \, dx$ where $R(x)$ is the distance from the curve to the axis of rotation (the radius of each disk), and $a$ and $b$ are the bounds of integration.

Here, we have two regions:

• The curve $y = x^2$ is the lower boundary, and
• The line $y = 4$ is the upper boundary.

The radius of the disk at any $x$ is the distance between $y = 4$ and $y = x^2$, i.e., $R(x) = 4 - x^2$.

Step 3: Set up the integral

Now we can set up the integral to find the volume:

$V = \pi \int_0^2 (4 - x^2)^2 \, dx$

This integral represents the volume of the solid generated by rotating the region between the curves about the $x$-axis.

Would you like me to solve this integral or clarify any part of the setup?

Relative Questions

1. How would the setup change if the region were rotated about the $y$-axis instead?
2. What would the integral look like if we used the shell method for this problem?
3. How can we modify the method if the region is rotated about the line $y = -2$ instead of the $x$-axis?
4. How do you approach finding the volume if $y = x^3$ replaces $y = x^2$ as the lower boundary?
5. What additional steps would be involved if the region were not confined to the first quadrant?

Tip

Always check the intersection points carefully when setting up integrals for volumes of revolution, as they determine the limits of integration.