Calculus Examples

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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 the integrand.
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Simplify each term.
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Rewrite as .
Expand using the FOIL Method.
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Apply the distributive property.
Apply the distributive property.
Apply the distributive property.
Simplify and combine like terms.
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Simplify each term.
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Multiply by by adding the exponents.
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Use the power rule to combine exponents.
Add and .
Rewrite using the commutative property of multiplication.
Multiply by by adding the exponents.
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Move .
Multiply by .
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Raise to the power of .
Use the power rule to combine exponents.
Add and .
Multiply by by adding the exponents.
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Move .
Multiply by .
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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.
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Move .
Multiply by .
Multiply by .
Subtract from .
Apply the distributive property.
Simplify.
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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
Simplify the answer.
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Combine and .
Substitute and simplify.
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Evaluate at and at .
Evaluate at and at .
Evaluate at and at .
Simplify.
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Raise to the power of .
Raising to any positive power yields .
Cancel the common factor of and .
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Factor out of .
Cancel the common factors.
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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 .
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Factor out of .
Cancel the common factors.
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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 .
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Factor out of .
Cancel the common factors.
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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.
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Multiply by .
Add and .
Raise to the power of .
Cancel the common factor of and .
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Factor out of .
Cancel the common factors.
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Factor out of .
Cancel the common factor.
Rewrite the expression.
Divide by .
Raising to any positive power yields .
Cancel the common factor of and .
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Factor out of .
Cancel the common factors.
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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.
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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
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