# Calculus Examples

,

Step 1

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

Step 2

Simplify each term.

Rewrite as .

Expand using the FOIL Method.

Apply the distributive property.

Apply the distributive property.

Apply the distributive property.

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

Multiply by by adding the exponents.

Move .

Use the power rule to combine exponents.

Add and .

Multiply by .

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 .

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 .

Rewrite using the commutative property of multiplication.

Multiply by by adding the exponents.

Move .

Multiply by .

Multiply by .

Subtract from .

Apply the product rule to .

Raise to the power of .

Multiply by .

Subtract from .

Step 3

Split the single integral into multiple integrals.

Step 4

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

Step 5

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

Step 6

Combine and .

Step 7

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

Step 8

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

Step 9

Combine and .

Step 10

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

Step 11

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

Step 12

Combine and .

Substitute and simplify.

Evaluate at and at .

Evaluate at and at .

Evaluate at and at .

Simplify.

Raise to the power of .

Raising to any positive power yields .

Cancel the common factor of and .

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 .

Raise to the power of .

Raising to any positive power yields .

Cancel the common factor of and .

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 .

Multiply by .

Multiply by .

Multiply by .

Multiply by .

Combine the numerators over the common denominator.

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 .

Cancel the common factor of and .

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 .

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 .

Multiply by .

Multiply by .

Multiply by .

Multiply by .

Combine the numerators over the common denominator.

Multiply by .

Multiply by .

Add and .

Combine and .

Multiply by .

Step 13

Divide by .

Step 14