Volume of Rotation: Disk Method

There are two ways to think about the disk method.

Method A: Slice and Stack

You can think of just taking the area of a circle and then stacking that circle horizontally:

• Slide the first radius dial: you'll see a yellow line extend from the x-axis to the function.

• Look at the face projection screen next to it: you see that this line becomes the radius of the circle. In the x-y plane, that is actually just the function itself.

• Scroll the second slider: this rotates that radius around to sweep out a full circle of area pi * r².

• Stack for volume: visualize stacking a bunch of these circular slices together from the start of the integral (a) to the end of the integral (b).

Method B: Revolve 2D Area

You can also think about it by first integrating the function itself (x² - 1) as you normally would, which gives you the 2D area underneath the curve:

• Revolve for volume: next, rotate that entire 2D area around the axis (multiplying by the circular circumference). You can see this by sliding the third slider.

• The formula: in the official format below, this is represented by the integral from a to b of f(x)² multiplied by pi. In the simulation to your right, f(x) is just the yellow radius line mapping from the x-axis to the function, representing the radius r.