# 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.

Remove unnecessary parentheses.

Simplify each term.

Use the power rule to combine exponents.

Add and .

Move .

Use the power rule to combine exponents.

Add and .

Move .

Use the power rule to combine exponents.

Add and .

Move .

Use the power rule to combine exponents.

Add and .

Move .

Use the power rule to combine exponents.

Add and .

Multiply by .

Move .

Use the power rule to combine exponents.

Add and .

Multiply by .

Move .

Use the power rule to combine exponents.

Add and .

Move .

Use the power rule to combine exponents.

Add and .

Multiply by .

Move .

Use the power rule to combine exponents.

Add and .

Multiply by .

Subtract from .

Add and .

Add and .

Subtract from .

Remove parentheses around .

Subtract from .

Since integration is linear, the integral of with respect to is .

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 , the integral of with respect to is .

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 , the integral of with respect to is .

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 , the integral of with respect to is .

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 , the integral of with respect to is .

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 .

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 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 factor.

Rewrite the expression.

Raising to any positive power yields .

Reduce the expression by cancelling 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 .

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 .

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 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 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 .

Multiply by .

Add and .

Raise to the power of .

Reduce the expression by cancelling 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 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 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 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 .

Multiply by .

Add and .

Write as a fraction with denominator .

Multiply and .

Move to the left of the expression .

Multiply by .