Question

Find the volume of the solid of revolution generated by revolving the region in the first quadrant bounded by the x-axis and the graph of
y = 4x − x2
about the following:
x axis:
y axis:
x=5:
y=4:

Answer 1

the volume of a shell is 2πrh so sketch the region and assign the values as needed.
about the x-axis, using the symmetry of the region
v = 2∫[0,2] 2πrh dy
where r = y and h = 2-x = 2-(2-√(4-y))
v = 2∫[0,4] 2πy(√(4-y)) dy = 512π/15
check, using discs
v = ∫[0,4] πr^2 dx
where r = y = 4x-x^2
v = ∫[0,4] π(4x-x^2)^2 dx = 512π/15

Now, you try the others. Post your work if you get stuck.