# 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.

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 and .

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

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

Combine and .

Substitute and simplify.

Evaluate at and at .

Evaluate at and at .

Raising to any positive power yields .

Reduce the expression by cancelling the common factors.

Factor out of .

Cancel the common factors.

Factor out of .

Cancel the common factor.

Rewrite the expression.

Divide by .

Multiply by .

Add and .

Raising to any positive power yields .

Reduce the expression by cancelling the common factors.

Factor out of .

Cancel the common factors.

Factor out of .

Cancel the common factor.

Rewrite the expression.

Divide by .

Multiply by .

Add and .

Simplify each term.

Simplify the numerator.

Apply the product rule to .

Raise to the power of .

Raise to the power of .

Multiply the numerator by the reciprocal of the denominator.

Cancel the common factor of .

Factor out the greatest common factor .

Cancel the common factor.

Rewrite the expression.

Multiply and .

Multiply .

Combine and .

Multiply by .

Reduce the expression by cancelling the common factors.

Factor out of .

Cancel the common factors.

Factor out of .

Cancel the common factor.

Rewrite the expression.

Move the negative in front of the fraction.

Simplify the numerator.

Apply the product rule to .

Raise to the power of .

Raise to the power of .

Multiply the numerator by the reciprocal of the denominator.

Multiply .

Multiply and .

Multiply by .

Multiply .

Combine 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 .

Combine.

Multiply by .

Combine the numerators over the common denominator.

Simplify the numerator.

Multiply by .

Add and .