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.
Multiply by by adding the exponents.
Use the power rule to combine exponents.
Add and .
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 by adding the exponents.
Move .
Multiply by .
Raise to the power of .
Use the power rule to combine exponents.
Add and .
Rewrite using the commutative property of multiplication.
Multiply by by adding the exponents.
Move .
Multiply by .
Multiply by .
Subtract from .
Apply the distributive property.
Simplify.
Multiply by .
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 .
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 .
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 .
To write as a fraction with a common denominator, multiply by .
Combine and .
Combine the numerators over the common denominator.
Simplify the numerator.
Multiply by .
Add and .
Raise to the power of .
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 .
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 .
Multiply by .
To write as a fraction with a common denominator, multiply by .
Combine and .
Combine the numerators over the common denominator.
Simplify the numerator.
Multiply by .
Subtract from .
Combine and .
Move to the left of .
Step 13
The result can be shown in multiple forms.
Exact Form:
Decimal Form:
Step 14