# Find the Area Under the Curve y=e^(2x) ; [0,4] ;
Substitute for into then solve for .
Replace with in the equation.
Solve the equation for .
Remove parentheses.
Graph each side of the equation. The solution is the x-value of the point of intersection.
No solution
No solution
No solution
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.
Integrate to find the area between and .
Combine the integrals into a single integral.
Subtract from .
Let . Then , so . Rewrite using and .
Let . Find .
Rewrite.
Divide by .
Substitute the lower limit in for in .
Multiply by .
Substitute the upper limit in for in .
Multiply by .
The values found for and will be used to evaluate the definite integral.
Rewrite the problem using , , and the new limits of integration.
Combine and .
Since is constant with respect to , move out of the integral.
The integral of with respect to is .
Combine and .
Substitute and simplify.
Evaluate at and at .
Simplify.
Anything raised to is .
Multiply by .
Simplify the numerator.
Rewrite as .
Rewrite as .
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Simplify.
Rewrite as .
Rewrite as .
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Simplify.
Rewrite as .
Since both terms are perfect squares, factor using the difference of squares formula, where and .
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