, ,

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.

Remove parentheses.

Split the single integral into multiple integrals.

The integral of with respect to is .

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

Let . Find .

Differentiate .

Differentiate using the chain rule, which states that is where and .

To apply the Chain Rule, set as .

Differentiate using the Exponential Rule which states that is where =.

Replace all occurrences of with .

Differentiate using the Power Rule which states that is where .

Move to the left of .

Reorder and .

Reorder and .

Substitute the lower limit in for in .

Simplify.

One to any power is one.

Simplify.

Substitute the upper limit in for in .

The values found for and will be used to evaluate the definite integral.

Rewrite the problem using , , and the new limits of integration.

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

Combine and .

Substitute and simplify.

Evaluate at and at .

Evaluate at and at .

Simplify.

Simplify each term.

Apply the distributive property.

Multiply .

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 each term.

Simplify the numerator.

Multiply by .

Add and .

Move the negative in front of the fraction.

Find the Area Between the Curves y=e^x , y=xe^(x^2) , (1,e)