Calculus Examples

,
Step 1
Set up the producer surplus where is the equilibrium quantity and is the equilibrium price.
Step 2
Evaluate the integral and simplify.
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Multiply by .
Since is constant with respect to , move out of the integral.
Multiply by .
By the Power Rule, the integral of with respect to is .
Simplify the answer.
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Combine and .
Substitute and simplify.
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Evaluate at and at .
Simplify.
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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 .
Subtract from .
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