To solve this problem, we need to compute the volume of the solid generated when the region enclosed by the curves $y = 2x$ and $y = x^2$ is rotated around the $x$-axis. We can do this using the method of disks or washers.

Step 1: Determine the points of intersection

First, we find where the two curves intersect by setting $y = 2x$ equal to $y = x^2$:

$2x = x^2$

Rearranging:

$x^2 - 2x = 0$ $x(x - 2) = 0$

So, the points of intersection are $x = 0$ and $x = 2$. These will be the limits of integration.

Step 2: Set up the washer method

For a rotation around the $x$-axis, the volume is given by the washer formula:

$V = \pi \int_{a}^{b} \left[ R_{\text{outer}}(x)^2 - R_{\text{inner}}(x)^2 \right] dx$

Here:

• $R_{\text{outer}}(x) = 2x$ (the curve farther from the x-axis, $y = 2x$)
• $R_{\text{inner}}(x) = x^2$ (the curve closer to the x-axis, $y = x^2$)

Thus, the volume integral becomes:

$V = \pi \int_{0}^{2} \left[ (2x)^2 - (x^2)^2 \right] dx$ $V = \pi \int_{0}^{2} \left( 4x^2 - x^4 \right) dx$

Step 3: Compute the integral

Now, we evaluate the integral:

$V = \pi \int_{0}^{2} \left( 4x^2 - x^4 \right) dx$

First, integrate term by term:

$\int 4x^2 \, dx = \frac{4x^3}{3}$ $\int x^4 \, dx = \frac{x^5}{5}$

Now apply the limits:

$V = \pi \left[ \frac{4x^3}{3} \Big|_0^2 - \frac{x^5}{5} \Big|_0^2 \right]$ $V = \pi \left[ \frac{4(2)^3}{3} - \frac{(2)^5}{5} \right]$ $V = \pi \left[ \frac{32}{3} - \frac{32}{5} \right]$

Step 4: Simplify

To simplify, find a common denominator for the fractions:

$V = \pi \left[ \frac{160}{15} - \frac{96}{15} \right]$ $V = \pi \cdot \frac{64}{15}$

Thus, the volume is:

$V = \frac{64\pi}{15}$

The volume of the solid generated by rotating the region around the x-axis is $\frac{64\pi}{15}$.

Would you like more details or have any questions?

Here are 5 questions related to this concept:

1. How would the volume change if the region were rotated around the y-axis instead?
2. What if we rotated around the line $y = -1$ instead of the x-axis?
3. Can you use the shell method to find the volume for this problem?
4. How would the integral change if one of the curves were shifted?
5. How do the limits of integration change if the region is bounded by different functions?

Tip: When dealing with volumes of revolution, always carefully identify the outer and inner functions before setting up the integral.