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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)

8.10 Volume with Disc Method: Revolving Around Other Axes

#disc

#volumeofrevolution

#volume

โฑ๏ธย ย 1 min read

written by

Anusha Tekumulla

anusha tekumulla

June 8, 2020


This topic is similar to topic 8.9 except we will be revolving our shape around other axes instead of just the x- or y-axis. This can get a bit confusing but weโ€™ll simplify it for you with an example problem.ย 

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

To keep it simple, weโ€™ll use the same y = โˆšx curve from the example in topic 8.9. This time, however, weโ€™ll revolve around the line y = 1 from x = 1 and x = 4.ย 

If you rotate this region around the line y = 1, the cross sections will be circles with radii (โˆšx) - 1. This is because our region is now shifted up one. When revolving around a line that is not the x- or y-axis, we must remember to take that into account when calculating the radius of our discs. With that, the area of each cross section will be ฯ€((โˆšx) - 1)2. This can be simplified to ฯ€(x - 2โˆšx + 1). Now we can integrate ฯ€(x - 2โˆšx + 1) from x = 1 and x = 4 to get the volume.ย 

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