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Unit 8

8.10 Volume with Disc Method: Revolving Around Other Axes

1 min readโ€ขjune 8, 2020

anusha

Anusha Tekumulla


AP Calculus AB/BCย โ™พ๏ธ

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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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