# 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

Apply the power rule and multiply exponents, .

Step 2.2

Multiply by .

Step 3

Split the single integral into multiple integrals.

Step 4

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

Step 5

Combine and .

Step 6

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

Step 7

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

Step 8

Step 8.1

Combine and .

Step 8.2

Substitute and simplify.

Step 8.2.1

Evaluate at and at .

Step 8.2.2

Evaluate at and at .

Step 8.2.3

Simplify.

Step 8.2.3.1

One to any power is one.

Step 8.2.3.2

Raising to any positive power yields .

Step 8.2.3.3

Cancel the common factor of and .

Step 8.2.3.3.1

Factor out of .

Step 8.2.3.3.2

Cancel the common factors.

Step 8.2.3.3.2.1

Factor out of .

Step 8.2.3.3.2.2

Cancel the common factor.

Step 8.2.3.3.2.3

Rewrite the expression.

Step 8.2.3.3.2.4

Divide by .

Step 8.2.3.4

Multiply by .

Step 8.2.3.5

Add and .

Step 8.2.3.6

One to any power is one.

Step 8.2.3.7

Raising to any positive power yields .

Step 8.2.3.8

Cancel the common factor of and .

Step 8.2.3.8.1

Factor out of .

Step 8.2.3.8.2

Cancel the common factors.

Step 8.2.3.8.2.1

Factor out of .

Step 8.2.3.8.2.2

Cancel the common factor.

Step 8.2.3.8.2.3

Rewrite the expression.

Step 8.2.3.8.2.4

Divide by .

Step 8.2.3.9

Multiply by .

Step 8.2.3.10

Add and .

Step 8.2.3.11

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

Step 8.2.3.12

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

Step 8.2.3.13

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

Step 8.2.3.13.1

Multiply by .

Step 8.2.3.13.2

Multiply by .

Step 8.2.3.13.3

Multiply by .

Step 8.2.3.13.4

Multiply by .

Step 8.2.3.14

Combine the numerators over the common denominator.

Step 8.2.3.15

Subtract from .

Step 8.2.3.16

Combine and .

Step 8.2.3.17

Move to the left of .

Step 9

The result can be shown in multiple forms.

Exact Form:

Decimal Form:

Step 10