# 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
Multiply the exponents in .
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
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
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