Question

Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the y-axis.
y = 8x − x^2, y = x

Answer 1

The curves intersect at (0,0) and (7,7)
So, using shells of thickness dx,
v = ∫[0,7] 2πrh dx
where r=x and h=(8x-x^2)-x = 7x-x^2
v = ∫[0,7] 2πx(7x-x^2) dx = 2401π/6

To use discs of thickness dy, you need to change the boundary at (7,7) so it gets a bit more complicated.

v1 = ∫[0,7] π(R^2-r^2) dy
where R = y and r = 4-√(16-y)
v1 = ∫[0,7] π(y^2-(4-√(16-y))^2) dy = 673π/6

v2 = ∫[7,16] π(R^2-r^2) dy
where R = 4+√(16-y) and r = 4-√(16-y)
v2 = ∫7,16] π*((4+√(16-y))^2-(4-√(16-y))^2) dy = 288π

so v = v1+v2 = 673π/6 + 1728π/6 = 2401π/6