Calculus Examples

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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
Simplify the integrand.
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Simplify each term.
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Remove parentheses.
Rewrite as .
Expand using the FOIL Method.
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Apply the distributive property.
Apply the distributive property.
Apply the distributive property.
Remove parentheses.
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 .
Multiply by by adding the exponents.
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Move .
Combine and
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Raise to the power of .
Use the power rule to combine exponents.
Add and .
Move to the left of the expression .
Multiply by .
Multiply by by adding the exponents.
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Move .
Combine and
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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 .
Combine and
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Raise to the power of .
Raise to the power of .
Use the power rule to combine exponents.
Add and .
Multiply by .
Subtract from .
Apply the distributive property.
Simplify.
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Multiply by .
Multiply by .
Simplify by adding terms.
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Remove unnecessary parentheses.
Subtract from .
Split the single integral into multiple integrals.
Since is constant with respect to , move out of the integral.
By the Power Rule, the integral of with respect to is .
Combine fractions.
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Write as a fraction with denominator .
Multiply and .
Since is constant with respect to , move out of the integral.
By the Power Rule, the integral of with respect to is .
Combine fractions.
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Write as a fraction with denominator .
Multiply and .
Since is constant with respect to , move out of the integral.
By the Power Rule, the integral of with respect to is .
Simplify the answer.
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Write as a fraction with denominator .
Multiply and .
Evaluate at and at .
Evaluate at and at .
Evaluate at and at .
Raise to the power of .
Raising to any positive power yields .
Reduce the expression by cancelling the common factors.
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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 .
Reduce the expression by cancelling the common factors.
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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 .
Write as a fraction with denominator .
Multiply and .
Multiply by .
Reduce the expression by cancelling the common factors.
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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 .
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Combine.
Multiply by .
Combine the numerators over the common denominator.
Simplify the numerator.
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Multiply by .
Add and .
Raise to the power of .
Reduce the expression by cancelling the common factors.
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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 .
Reduce the expression by cancelling the common factors.
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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 .
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Combine.
Multiply by .
Combine the numerators over the common denominator.
Simplify the numerator.
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Multiply by .
Subtract from .
Write as a fraction with denominator .
Multiply and .
Move to the left of the expression .
Multiply by .
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