# 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 to get .

Move .

Use the power rule to combine exponents.

Add and to get .

Move .

Use the power rule to combine exponents.

Add and to get .

Move .

Use the power rule to combine exponents.

Add and to get .

Move .

Use the power rule to combine exponents.

Add and to get .

Multiply by to get .

Move .

Use the power rule to combine exponents.

Add and to get .

Multiply by to get .

Move .

Use the power rule to combine exponents.

Add and to get .

Move .

Use the power rule to combine exponents.

Add and to get .

Multiply by to get .

Move .

Use the power rule to combine exponents.

Add and to get .

Multiply by to get .

Subtract from to get .

Add and to get .

Add and to get .

Subtract from to get .

Remove parentheses around .

Subtract from to get .

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 to get .

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 to get .

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 to get .

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 to get .

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 to get .

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 to get .

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 to get .

Multiply by to get .

Add and to get .

Raise to the power of to get .

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 to get .

Multiply by to get .

Add and to get .

Write as a fraction with denominator .

Multiply and to get .

Multiply by to get .

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 to get .

Combine.

Multiply by to get .

Combine the numerators over the common denominator.

Multiply by to get .

Multiply by to get .

Multiply by to get .

Subtract from to get .

Move the negative in front of the fraction.

Raise to the power of to get .

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 to get .

Multiply by to get .

Add and to get .

Write as a fraction with denominator .

Multiply and to get .

Multiply by to get .

Reduce the expression by cancelling the common factors.

Factor out of .

Cancel the common factor.

Rewrite the expression.

Divide by to get .

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 to get .

Combine the numerators over the common denominator.

Multiply by to get .

Multiply by to get .

Add and to get .

Raise to the power of to get .

Reduce the expression by cancelling the common factors.

Factor out of .

Cancel the common factor.

Rewrite the expression.

Divide by to get .

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 to get .

Multiply by to get .

Add and to get .

Multiply by to get .

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 to get .

Combine the numerators over the common denominator.

Multiply by to get .

Subtract from to get .

Move the negative in front of the fraction.

Raise to the power of to get .

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 to get .

Multiply by to get .

Add and to get .

Write as a fraction with denominator .

Multiply and to get .

Multiply by to get .

Reduce the expression by cancelling the common factors.

Factor out of .

Cancel the common factor.

Rewrite the expression.

Divide by to get .

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

Combine.

Multiply by to get .

Combine the numerators over the common denominator.

Multiply by to get .

Multiply by to get .

Add and to get .

Write as a fraction with denominator .

Multiply and to get .

Move to the left of the expression .

Reorder terms.