Revolve the Enclosed Region Calculus

Question

A triangle has corner points (1,1), (1,2), (2,2). Revolve the enclosed region about the y-axis and find the volume.

Answer 1

Using the washer method, revolve a thin horizontal slice (of width dy) of the triangular region about the y-axis.

We obtain a washer with outer radius y and inner radius 1.

Volume of washer = πy²dy - π(1)²dy = π(y²-1)dy

Volume of solid = π∫_{(from 1 to 2)}(y² - 1)dy = [(1/3)y³ - y]_{(from 1 to 2)}

= (8/3 - 2) - (1/3 - 1) = 4/3