# 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

Step 2.1

Simplify each term.

Step 2.1.1

Rewrite as .

Step 2.1.2

Expand using the FOIL Method.

Step 2.1.2.1

Apply the distributive property.

Step 2.1.2.2

Apply the distributive property.

Step 2.1.2.3

Apply the distributive property.

Step 2.1.3

Simplify and combine like terms.

Step 2.1.3.1

Simplify each term.

Step 2.1.3.1.1

Rewrite using the commutative property of multiplication.

Step 2.1.3.1.2

Multiply by by adding the exponents.

Step 2.1.3.1.2.1

Move .

Step 2.1.3.1.2.2

Use the power rule to combine exponents.

Step 2.1.3.1.2.3

Add and .

Step 2.1.3.1.3

Multiply by .

Step 2.1.3.1.4

Rewrite using the commutative property of multiplication.

Step 2.1.3.1.5

Multiply by by adding the exponents.

Step 2.1.3.1.5.1

Move .

Step 2.1.3.1.5.2

Multiply by .

Step 2.1.3.1.5.2.1

Raise to the power of .

Step 2.1.3.1.5.2.2

Use the power rule to combine exponents.

Step 2.1.3.1.5.3

Add and .

Step 2.1.3.1.6

Multiply by .

Step 2.1.3.1.7

Rewrite using the commutative property of multiplication.

Step 2.1.3.1.8

Multiply by by adding the exponents.

Step 2.1.3.1.8.1

Move .

Step 2.1.3.1.8.2

Multiply by .

Step 2.1.3.1.8.2.1

Raise to the power of .

Step 2.1.3.1.8.2.2

Use the power rule to combine exponents.

Step 2.1.3.1.8.3

Add and .

Step 2.1.3.1.9

Multiply by .

Step 2.1.3.1.10

Rewrite using the commutative property of multiplication.

Step 2.1.3.1.11

Multiply by by adding the exponents.

Step 2.1.3.1.11.1

Move .

Step 2.1.3.1.11.2

Multiply by .

Step 2.1.3.1.12

Multiply by .

Step 2.1.3.2

Subtract from .

Step 2.1.4

Apply the product rule to .

Step 2.1.5

Raise to the power of .

Step 2.1.6

Multiply by .

Step 2.2

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

Step 12.1

Combine and .

Step 12.2

Substitute and simplify.

Step 12.2.1

Evaluate at and at .

Step 12.2.2

Evaluate at and at .

Step 12.2.3

Evaluate at and at .

Step 12.2.4

Simplify.

Step 12.2.4.1

Raise to the power of .

Step 12.2.4.2

Raising to any positive power yields .

Step 12.2.4.3

Cancel the common factor of and .

Step 12.2.4.3.1

Factor out of .

Step 12.2.4.3.2

Cancel the common factors.

Step 12.2.4.3.2.1

Factor out of .

Step 12.2.4.3.2.2

Cancel the common factor.

Step 12.2.4.3.2.3

Rewrite the expression.

Step 12.2.4.3.2.4

Divide by .

Step 12.2.4.4

Multiply by .

Step 12.2.4.5

Add and .

Step 12.2.4.6

Combine and .

Step 12.2.4.7

Multiply by .

Step 12.2.4.8

Raise to the power of .

Step 12.2.4.9

Raising to any positive power yields .

Step 12.2.4.10

Cancel the common factor of and .

Step 12.2.4.10.1

Factor out of .

Step 12.2.4.10.2

Cancel the common factors.

Step 12.2.4.10.2.1

Factor out of .

Step 12.2.4.10.2.2

Cancel the common factor.

Step 12.2.4.10.2.3

Rewrite the expression.

Step 12.2.4.10.2.4

Divide by .

Step 12.2.4.11

Multiply by .

Step 12.2.4.12

Add and .

Step 12.2.4.13

Combine and .

Step 12.2.4.14

Multiply by .

Step 12.2.4.15

Move the negative in front of the fraction.

Step 12.2.4.16

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

Step 12.2.4.17

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

Step 12.2.4.18

Write each expression with a common denominator of , by multiplying each by an appropriate factor of .

Step 12.2.4.18.1

Multiply by .

Step 12.2.4.18.2

Multiply by .

Step 12.2.4.18.3

Multiply by .

Step 12.2.4.18.4

Multiply by .

Step 12.2.4.19

Combine the numerators over the common denominator.

Step 12.2.4.20

Multiply by .

Step 12.2.4.21

Multiply by .

Step 12.2.4.22

Subtract from .

Step 12.2.4.23

Move the negative in front of the fraction.

Step 12.2.4.24

Raise to the power of .

Step 12.2.4.25

Raising to any positive power yields .

Step 12.2.4.26

Cancel the common factor of and .

Step 12.2.4.26.1

Factor out of .

Step 12.2.4.26.2

Cancel the common factors.

Step 12.2.4.26.2.1

Factor out of .

Step 12.2.4.26.2.2

Cancel the common factor.

Step 12.2.4.26.2.3

Rewrite the expression.

Step 12.2.4.26.2.4

Divide by .

Step 12.2.4.27

Multiply by .

Step 12.2.4.28

Add and .

Step 12.2.4.29

Combine and .

Step 12.2.4.30

Multiply by .

Step 12.2.4.31

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

Step 12.2.4.32

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

Step 12.2.4.33

Write each expression with a common denominator of , by multiplying each by an appropriate factor of .

Step 12.2.4.33.1

Multiply by .

Step 12.2.4.33.2

Multiply by .

Step 12.2.4.33.3

Multiply by .

Step 12.2.4.33.4

Multiply by .

Step 12.2.4.34

Combine the numerators over the common denominator.

Step 12.2.4.35

Multiply by .

Step 12.2.4.36

Multiply by .

Step 12.2.4.37

Add and .

Step 12.2.4.38

Combine and .

Step 12.2.4.39

Multiply by .

Step 13

Divide by .

Step 14