# 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 .

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 .

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 .

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 .

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 .

Combine.

Multiply by .

Combine the numerators over the common denominator.

Simplify the numerator.

Multiply by .

Add and .

Raise to the power of .

Reduce the expression by cancelling the common factors.

Factor out of .

Cancel the common factor.

Rewrite the expression.

Divide by .

Multiply by .

Subtract from .

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 .

Combine.

Multiply by .

Combine the numerators over the common denominator.

Simplify the numerator.

Multiply by .

Add and .

Raise to the power of .

Reduce the expression by cancelling the common factors.

Factor out of .

Cancel the common factor.

Rewrite the expression.

Divide by .

Raise to the power of .

Move the negative in front of the fraction.

Multiply by .

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 .

Combine.

Multiply by .

Combine the numerators over the common denominator.

Simplify the numerator.

Multiply by .

Add and .

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 .

Combine.

Multiply by .

Combine the numerators over the common denominator.

Simplify the numerator.

Multiply by .

Multiply by .

Multiply by .

Subtract from .