5. Find the volume of a cone with radius
by using the shell method on the appropriate region which, when rotated around the
-axis, produces a cone with the given characteristics.
You could set up the appropriate region in any of the four quadrants. Here we set it up in the first quadrant. Since we are revolving around the -axis, the direction will be the height and the radius will be along the direction. So we need a line that passes through the points and . The slope of this line is
and the -intercept is . Thus, the equation of the line is
The -values of the region run from to . Since the function is positive throughout the region we can drop the absolute value sign. The volume will be