# 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.

Simplify .

Use the power rule to combine exponents.

Add and .

Move .

Simplify .

Use the power rule to combine exponents.

Add and .

Move .

Simplify .

Use the power rule to combine exponents.

Add and .

Move .

Simplify .

Use the power rule to combine exponents.

Add and .

Move .

Simplify .

Use the power rule to combine exponents.

Add and .

Multiply by .

Move .

Simplify .

Use the power rule to combine exponents.

Add and .

Multiply by .

Move .

Simplify .

Use the power rule to combine exponents.

Add and .

Move .

Simplify .

Use the power rule to combine exponents.

Add and .

Multiply by .

Move .

Simplify .

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 .