Question

How to calculate the volume of rotating object formed by the area between curve y=x^2 and y=3x which is rotating through :
a. x axis
b. y axis ; using skin tube methode

Answer 1

The curves intersect at (0,0) and (3,9)

So, rotating around the x-axis, using

discs:
v = ∫[0,3] π(R^2-r^2) dx
where R=3x and r=x^2
v = ∫[0,3] π((3x)^2-(x^2)^2) dx = 162π/5

shells (tubes):
v = ∫[0,3] 2πrh dy
where r=y and h=(√y)-(y/3)
v = ∫[0,9] 2πy(√y-y/3) dy = 162π/5

Now do the other part, interchanging the x and y interpretations.