Question

Find the volume of the solid generated by revolving the triangular region with vertices (1,1), (b,1), and (1,h) about:
a) the x-axis
b) the y-axis

Answer 1

If a<1 and b<1 we get
Using discs:
pi*Int(R^2 - r^2) dx [b,1]
R = 1
r = y = (a-1)(x-b)/(1-b) + 1

= pi/3 (a-1)(a+2)(b-1)

Using shells:
2pi*Int(rh)dy [a,1]
r = y
h = 1-x = 1 - [(1-b)(y-1)/(a-1) + b]

= pi/3 (a-1)(a+2)(b-1)

If a or b > 1, then change integration limits and (1-a) -> (a-1), but the answer is the same

Thank you, Wolframalpha!

Answer 2

oh - that's just around the x-axis. Good luck on the y-axis. By symmetry, the results should look quite similar.