Question

Use cylindrical shells to find the volume V of the solid.
A right circular cone with height 9h and base radius 5r. The answer is 75πhr^2 but my answer is 39πhr^2. How???

Answer 1

forget the 9h and 5r and just call them 9 and 5. The r^2h factor will come out at the end.
If you take a side view, where the axis of the cone lies on the x-axis, then the line of the side is x = 9 - 9/5 y

So, to rotate that around the x-axis,
v = ∫[0,5] 2πrh dy
where r=y and h=x
v = ∫[0,5] 2πy(9 - 9/5 y) dy = 18π∫[0,5] y - y^2/5) dy
= 18π(y^2/2 - y^3/15)[0,5] = 18π(25/2 - 25/3) = 18π*25/6 = 75π

How you got 39π I don't know, since you didn't see fit to show your work . . .