While integrals can be used to find area between curves, they can also be used to find the volume of curves after being rotated around a central axis. This axis is known as the axis of rotation, for obvious reasons. There are a few methods to finding volume: disk/washer method and shell method. The example below will show the disk method in vivid detail.

Write your equtaion

This line is going to be rotated about the x-axis, between this line and the x-axis from x=0 to x=1.

Create a crudely Drawn picture

As Mr. Moyer advised (and sometimes) required on all problems, draw a bad picture to help with knowing how to set up your integral.

write the standard volume equation

The disk method uses circular cross sections that are sliced very thin and the summed to get the overall volume, so we have this equation for volume.

Substitute in for your area

Because the cross section in the disk method is circular, the area function will be A=πr^2.

insert equation and bounds

In the disk method, your function is going to be your radius, since the function is what dictates how far away from the x-axis the line is. The bounds were given to you, so you can insert those into the spot of a and b. Then, substitute your function in for r. Because π is a constant, we can move it outside the integral sign to avoid any confusion when integrating. Do not forget about the exponent outside the function, or you will fail and die.

Simplify

Because the function is just simply the square root of x, we can just drop the square root sign. However, if the function happened to be a polynomial, you would have to expand it fully in order to be able to integrate it.

Integrate

Do a simple integration on x, and you should get x^2/2. Your dx drops off and you can then get the value from your bounds.

Solve using bounds

The bounds were x=0 and x=1, so we do π(1^2/2-0^2) because the value of an integral is determined using (fb-fa).

Simplify

The rest is just basic algebra. Subtract the 0 from 1/2 and then multiply by π. Your volume should be π/2.

The axis of rotation can also be around the y-axis. Follow this link for a detailed look into rotation around the y-axis. https://www.khanacademy.org/math/integral-calculus/solid_revolution_topic/disc-method/v/disc-method-around-y-axis