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ap calc

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👑 Unit 1: Limits & Continuity

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#disc

#volumeofrevolution

#volume

⏱️ **1 min read**

written by

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 we’ll simplify it for you with an example problem.

To keep it simple, we’ll use the same y = √x curve from the example in topic 8.9. This time, however, we’ll revolve around the line y = 1 from x = 1 and x = 4.

If you **rotate** this region around the line y = 1, the cross sections will be circles with radii (√x) - 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 π((√x) - 1)2. This can be **simplified **to π(x - 2√x + 1). Now we can **integrate **π(x - 2√x + 1) from x = 1 and x = 4 to get the volume.

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