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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8.10Volume with Disc Method: Revolving Around Other Axes