# Calculus Examples

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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.

Apply the distributive property.

Multiply by to get .

Remove unnecessary parentheses.

Since integration is linear, the integral of with respect to is .

By the Power Rule, the integral of with respect to is .

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 .

Write as a fraction with denominator .

Multiply and to get .

Since is constant with respect to , the integral of with respect to is .

Combine and .

Evaluate at and at .

Evaluate at and at .

Raise to the power of 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 .

Combine.

Multiply by to get .

Combine the numerators over the common denominator.

Multiply by to get .

Add and to get .

Raise to the power of to get .

Reduce the expression by cancelling the common factors.

Factor out of .

Cancel the common factor.

Rewrite the expression.

Divide by to get .

Multiply by to get .

Subtract from 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 .

Combine.

Multiply by to get .

Combine the numerators over the common denominator.

Multiply by to get .

Add and to get .

Raise to the power of to get .

Reduce the expression by cancelling the common factors.

Factor out of .

Cancel the common factor.

Rewrite the expression.

Divide by to get .

Raise to the power of to get .

Move the negative in front of the fraction.

Multiply by 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 .

Combine.

Multiply by to get .

Combine the numerators over the common denominator.

Multiply by to get .

Add and to get .

To write as a fraction with a common denominator, multiply by .

To write as a fraction with a common denominator, multiply by .

Combine.

Multiply by to get .

Combine.

Multiply by to get .

Combine the numerators over the common denominator.

Multiply by to get .

Multiply by to get .

Multiply by to get .

Reorder terms.