ap calc study guides

๐Ÿ‘‘ย  Unit 1: Limits & Continuity

๐Ÿถย  Unit 8: Applications of Integration

โ™พย  Unit 10: Infinite Sequences and Series (BC Only)

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

#disc

#volumeofrevolution

#volume

โฑ๏ธย ย 1 min read

written by

Anusha Tekumulla

anusha tekumulla

June 8, 2020


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.ย 

๐Ÿ” Example Problem: Finding the Volume Using the Disc Method

Letโ€™s look at the region between the curve y = โˆšx and the x-axis from x = 0 and x = 1.ย 

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(667).png?alt=media&token=25c6ddb1-bb19-47d9-a4a4-9823e9c4d2dc

If you rotate this region around the x-axis, the cross sections will be circles with radii โˆšx. Thus, the area of each cross section will be ฯ€(โˆšx)^2 or ฯ€x. Now we can integrate ฯ€x from x = 0 and x = 1 to get the volume.ย 

Now, letโ€™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 beย ย 

ฯ€[f(x)]2

To find the volume, evaluate the integral.ย 

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(669).png?alt=media&token=199618bc-bb38-441f-bf2e-ee9d51f61c32

Now, you try to use the formula with this example problem:ย 

๐Ÿ” 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?ย 

Answer: 58.643

Solution:ย In this example, your function is y = x + 2 which looks like this:ย 

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(671).png?alt=media&token=6f655e49-ae32-4021-b74d-adeccc1f9ba7

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