Complex calculus problem involving derivatives and integrals

Full Answer Section

In this case, the radius of each disc is equal to the distance between the curve y=x2 and the $x$-axis. To find this distance, we need to solve the equation y=x2 for $x$.

x = sqrt(y)

Therefore, the radius of each disc is given by the equation r=y​.

Now that we have the equation for the radius of each disc, we can find the equation for the area of each disc.

A = \pi (\sqrt{y})^2 = \pi y

Now, we can use the disc method to find the volume of the solid. The volume of the solid is given by the following integral:

V = \int_a^b \pi y dx

where $a$ and $b$ are the $x$-coordinates of the endpoints of the region that we are revolving.

In this case, the $x$-coordinates of the endpoints of the region are $x=0$ and $x=2$. Therefore, the integral that gives the volume of the solid is as follows:

V = \int_0^2 \pi y dx

Now, we need to evaluate the integral.

V = \int_0^2 \pi (x^2) dx = \int_0^2 \pi x^2 dx

We can evaluate this integral using the following formula:

\int x^n dx = \frac{x^{n+1}}{n+1} + C

where $C$ is an arbitrary constant of integration.

V = \left[ \frac{\pi x^3}{3} \right]_0^2 = \frac{\pi (2)^3}{3} - \frac{\pi (0)}{3} = \frac{8 \pi}{3}

Therefore, the volume of the solid is 38π​.

Steps taken to arrive at the solution:

1. Identify the method of calculus that will be used to solve the problem. In this case, we will use the disc method.
2. Set up the integral that gives the volume of the solid.
3. Evaluate the integral.
4. Interpret the result.

I hope this explanation is helpful. Please let me know if you have any other questions.

Sample Answer

Find the volume of the solid formed by revolving the region bounded by the curves y=x2 and $y=2x$ about the $x$-axis.

Solution:

To solve this problem, we will use the disc method. The disc method is a calculus method for finding the volume of a solid of revolution. It works by dividing the solid into infinitely thin discs and then summing the volumes of the discs.

To use the disc method, we first need to find the equation for the area of each disc. The area of a disc is given by the formula πr2, where $r$ is the radius of the disc.