Calculus Examples

,
Reorder and .
The area of the region between the curves is defined as the integral of the upper curve minus the integral of the lower curve over each region. The regions are determined by the intersection points of the curves. This can be done algebraically or graphically.
Combine the integrals into a single integral.
Multiply by to get .
Subtract from to get .
Since integration is linear, the integral of with respect to is .
Since is constant with respect to , the integral of with respect to is .
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 to get .
Since is constant with respect to , the integral of with respect to is .
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 to get .
Evaluate at and at .
Evaluate at and at .
Raising to any positive power yields .
Reduce the expression by cancelling the common factors.
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Factor out of .
Cancel the common factor.
Rewrite the expression.
Divide by to get .
Multiply by to get .
Add and to get .
Raising to any positive power yields .
Reduce the expression by cancelling the common factors.
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Factor out of .
Cancel the common factor.
Rewrite the expression.
Divide by to get .
Multiply by to get .
Add and to get .
Simplify each term.
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Simplify the numerator.
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Apply the product rule to .
Raise to the power of to get .
Raise to the power of to get .
Multiply the numerator by the reciprocal of the denominator.
Cancel the common factor of .
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Factor out the greatest common factor .
Cancel the common factor.
Rewrite the expression.
Multiply and to get .
Simplify .
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Write as a fraction with denominator .
Multiply and to get .
Multiply by to get .
Reduce the expression by cancelling the common factors.
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Factor out of .
Cancel the common factor.
Rewrite the expression.
Move the negative in front of the fraction.
Simplify .
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Write as a fraction with denominator .
Multiply and to get .
Evaluate.
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Simplify each term.
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Simplify the numerator.
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Apply the product rule to .
Raise to the power of to get .
Raise to the power of to get .
Simplify the numerator.
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Write as a fraction with denominator .
Multiply and to get .
Multiply the numerator by the reciprocal of the denominator.
Multiply by to get .
Simplify .
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Multiply and to get .
Multiply by to get .
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 to get .
Combine the numerators over the common denominator.
Simplify the numerator.
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Multiply by to get .
Multiply by to get .
Add and .
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