Question

1. Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the line x = 6.
y = x, y = 0, y = 5, x = 6

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

3. The region bounded by the given curves is rotated about the specified axis. Find the volume V of the resulting solid by any method.
x = (y − 6)^2, x = 1; about y = 5

Answer 1

#1.using discs of thickness dy,
v = ∫[0,5] πr^2 dy
where r = 6-x = 6-y
v = ∫[0,5] π(6-y)^2 dy = 215π/3

using shells of thickness dx,
v1 = ∫[0,5] 2πrh dx
where r = 6-x and h = y = x
v1 = ∫[0,5] 2π(6-x)x dx = 200π/3

v2 is just a cylinder or radius 1 and height 5, or 5π
so the total volume is
v = v1 + v2 = 215π/3

Take a crack at the others. Post your work if you get stuck
Be sure to sketch the region, and watch for boundary changes.