, ,

To find the volume of the solid, first define the area of each slice then integrate across the range. The area of each slice is the area of a circle with radius and .

where

Apply the product rule to .

Raise to the power of .

Multiply the exponents in .

Apply the power rule and multiply exponents, .

Multiply by .

Since is constant with respect to , move out of the integral.

Move to the left of .

By the Power Rule, the integral of with respect to is .

Combine and .

Substitute and simplify.

Evaluate at and at .

Simplify.

One to any power is one.

Raising to any positive power yields .

Cancel the common factor of and .

Factor out of .

Cancel the common factors.

Factor out of .

Cancel the common factor.

Rewrite the expression.

Divide by .

Multiply by .

Add and .

Combine and .

Combine and .

Find the Volume y=8x^3 , y=0 , x=1