# Calculus Examples

,

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 and

Simplify each term.

Remove parentheses.

Rewrite as .

Expand using the FOIL Method.

Apply the distributive property.

Apply the distributive property.

Apply the distributive property.

Remove parentheses.

Simplify and combine like terms.

Simplify each term.

Multiply by by adding the exponents.

Use the power rule to combine exponents.

Add and .

Multiply by by adding the exponents.

Move .

Combine and

Raise to the power of .

Use the power rule to combine exponents.

Add and .

Move to the left of the expression .

Multiply by .

Multiply by by adding the exponents.

Move .

Combine and

Raise to the power of .

Use the power rule to combine exponents.

Add and .

Multiply by by adding the exponents.

Move .

Combine and

Raise to the power of .

Raise to the power of .

Use the power rule to combine exponents.

Add and .

Multiply by .

Subtract from .

Apply the distributive property.

Simplify.

Multiply by .

Multiply by .

Simplify by adding terms.

Remove unnecessary parentheses.

Subtract from .

Split the single integral into multiple integrals.

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

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

Write as a fraction with denominator .

Multiply and .

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

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

Write as a fraction with denominator .

Multiply and .

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

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

Write as a fraction with denominator .

Multiply and .

Evaluate at and at .

Evaluate at and at .

Evaluate at and at .

Raise to the power of .

Raising to any positive power yields .

Reduce the expression by cancelling the common factors.

Factor out of .

Cancel the common factors.

Factor out of .

Cancel the common factor.

Rewrite the expression.

Divide by .

Multiply by .

Add and .

Raise to the power of .

Raising to any positive power yields .

Reduce the expression by cancelling the common factors.

Factor out of .

Cancel the common factors.

Factor out of .

Cancel the common factor.

Rewrite the expression.

Divide by .

Multiply by .

Add and .

Write as a fraction with denominator .

Multiply and .

Multiply by .

Reduce the expression by cancelling the common factors.

Factor out of .

Cancel the common factors.

Factor out of .

Cancel the common factor.

Rewrite the expression.

Divide by .

To write as a fraction with a common denominator, multiply by .

Write each expression with a common denominator of , by multiplying each by an appropriate factor of .

Combine.

Multiply by .

Combine the numerators over the common denominator.

Simplify the numerator.

Multiply by .

Add and .

Raise to the power of .

Reduce the expression by cancelling the common factors.

Factor out of .

Cancel the common factors.

Factor out of .

Cancel the common factor.

Rewrite the expression.

Divide by .

Raising to any positive power yields .

Reduce the expression by cancelling the common factors.

Factor out of .

Cancel the common factors.

Factor out of .

Cancel the common factor.

Rewrite the expression.

Divide by .

Multiply by .

Add and .

Multiply by .

To write as a fraction with a common denominator, multiply by .

Write each expression with a common denominator of , by multiplying each by an appropriate factor of .

Combine.

Multiply by .

Combine the numerators over the common denominator.

Simplify the numerator.

Multiply by .

Subtract from .

Write as a fraction with denominator .

Multiply and .

Move to the left of the expression .

Multiply by .