Question

A triangular region is defined by the vertices (1, 1), (2, 2), and (5, 2). If a solid is generated by revolving the region about the y-axis, what is the volume of the solid?

Answer 1

Consider the line passing through the points (1,1) and (2,2) - this line would have an equation y=x --> x=y.
Consider the line passing through the points (1,1) and (5,2) - this line would have an equation y=0.25x+0.75 --> x=4y-3.
To find the volume of the resultant solid, we simply need to evaluate the following integral:

\[\pi[\int\limits_{1}^{2}(4y-3)^{2} - y^{2}dy]\]