# 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.

Rewrite as .

Expand by multiplying each term in the first expression by each term in the second expression.

Simplify each term.

Multiply by by adding the exponents.

Use the power rule to combine exponents.

Add and .

Rewrite using the commutative property of multiplication.

Multiply by by adding the exponents.

Move .

Use the power rule to combine exponents.

Add and .

Rewrite using the commutative property of multiplication.

Multiply by by adding the exponents.

Move .

Multiply by .

Raise to the power of .

Use the power rule to combine exponents.

Add and .

Multiply by by adding the exponents.

Move .

Use the power rule to combine exponents.

Add and .

Rewrite using the commutative property of multiplication.

Multiply by by adding the exponents.

Use the power rule to combine exponents.

Add and .

Multiply by .

Rewrite using the commutative property of multiplication.

Multiply by by adding the exponents.

Multiply by .

Raise to the power of .

Use the power rule to combine exponents.

Add and .

Multiply by .

Multiply by by adding the exponents.

Move .

Multiply by .

Raise to the power of .

Use the power rule to combine exponents.

Add and .

Rewrite using the commutative property of multiplication.

Multiply by by adding the exponents.

Multiply by .

Raise to the power of .

Use the power rule to combine exponents.

Add and .

Multiply by .

Multiply by .

Multiply by .

Subtract from .

Add and .

Add and .

Subtract from .

Subtract from .

Split the single integral into multiple integrals.

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

Combine and .

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

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

Combine and .

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

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

Combine and .

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

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

Combine and .

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

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

Combine and .

Substitute and simplify.

Evaluate at and at .

Evaluate at and at .

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 .

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.

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 .

Combine and .

Multiply by .

Move the negative in front of the fraction.

To write as a fraction with a common denominator, 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.

Multiply by .

Combine the numerators over the common denominator.

Simplify the numerator.

Multiply by .

Multiply by .

Subtract from .

Move the negative in front of the fraction.

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 .

Combine 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 .

Move the negative in front of the fraction.

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 .

Combine 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 .

Combine.

Multiply by .

Combine the numerators over the common denominator.

Simplify the numerator.

Multiply by .

Add and .

Combine and .

Move to the left of .