Question

Volume problem:

y=x^3, x=y^3; rotated about x-axis

I first made y=x^1/3, then x^3=x^1/3

Then I tried to do an integration from 0 to 1 with x^1/3-x^3, but I can't get one of the possible answers, which are:

A. 16/35π
B. 16/7π
C. 18/35
D. 7/2
E. 16π

Any advice?

Answer 1

| = integrate symbol

y = x^3
y^3 = x, y = x^(1/3)

Using the formula
pi | (f(x))^2 - (g(x))^2 dx

pi | (x^3)^2 - x^(1/3)^2 dx
pi | (x^6) - x^(2/3) dx
pi ( 1/7 x^7 - 3/5 x^(5/3))

When I evaluated from 0 to 1,
I got,
- 16/35 pi

Answer 2

Correction for above,

| = integrate symbol

y = x^3
y^3 = x, y = x^(1/3)

Using the formula
pi | (f(x))^2 - (g(x))^2 dx

pi | x^(1/3)^2 - (x^3)^2 dx
pi | x^(2/3) - (x^6) dx
pi ( 3/5 x^(5/3) - 1/7 x^7 )

When I evaluated from 0 to 1,
I got,
16/35 pi