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Unit 8

# 8.9 Volume with Disc Method: Revolving Around the x- or y-Axis

Anusha Tekumulla

### AP Calculus AB/BCΒ βΎοΈ

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The disc method is a way to find the volume by rotating around the x- or y-axis. In this situation, we will find the volume by adding up a bunch of infinitely thin circles.Β

### π Example Problem: Finding the Volume Using the Disc Method

Letβs look at the region between the curve y = βx and the x-axis from x = 0 and x = 1.Β
If you rotate this region around the x-axis, the cross sections will be circles with radii βx. Thus, the area of each cross section will be Ο(βx)^2 or Οx. Now we can integrate Οx from x = 0 and x = 1 to get the volume.Β
Now, letβs generalize this. If you have a region whose area is bounded by the curve y = f(x) and the x-axis on the interval [a,b], each disk has a radius of f(x), and the area of the disk will beΒ Β
Ο[f(x)]2
To find the volume, evaluate the integral.Β
Now, you try to use the formula with this example problem:Β

### π Example Problem: Finding the Volume Using the Disc Method #2

If rotate the function y = x + 2 about the x-axis from x = 0 to x = 2, what is the volume of the figure?Β
Solution:Β In this example, your function is y = x + 2 which looks like this:Β

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