If you choose cylindrical shells, slice the region vertically; you’ll need to adjust for the axis of revolution.
Each radius can be found by subtracting x from 8. (Not 8 from x. That was the tricky part, in case you
missed it.) The integral to evaluate is
2 π (8 − x)[(16 − x^2 ) − (16 − 4x)] dxIf you choose washers, slice the region horizontally. The radius of each washer is found by subtracting
each equation from 8. Notice also that the curve x = is now the outer radius of the washer, and the
curve x = is the inner radius. The integral looks like the following:
PROBLEM 7. Find the volume of a solid whose base is the region between the x-axis and the curve y = 4 −
x^2 , and whose cross-sections perpendicular to the x-axis are equilateral triangles with a side that lies on
the base.
Answer: The curve y = 4 − x^2 intersects the x-axis at x = −2 and x = 2. The side of the triangle is 4 − x^2 ,
so all that we have to do is evaluate .