1 min readβ’june 8, 2020

Anusha Tekumulla

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 weβll simplify it for you with an example problem.Β

To keep it simple, weβll use the same y = βx curve from the example in topic 8.9. This time, however, weβll revolve around the line y = 1 from x = 1 and x = 4.Β

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

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