# Find the Area Between the Curves y=2x-x^2 , y=2x-4

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Solve by substitution to find the intersection between the curves.
Substitute for into then solve for .
Replace with in the equation.
Solve the equation for .
Set the equation equal to zero.
Move all the expressions to the left side of the equation.
Move to the left side of the equation by subtracting it from both sides.
Move to the left side of the equation by adding it to both sides.
Combine the opposite terms in .
Subtract from .
Factor the left side of the equation.
Rewrite as .
Reorder and .
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Replace the left side with the factored expression.
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Set the first factor equal to and solve.
Set the first factor equal to .
Subtract from both sides of the equation.
Set the next factor equal to and solve.
Set the next factor equal to .
Subtract from both sides of the equation.
Multiply each term in by
Multiply each term in by .
Multiply .
Multiply by .
Multiply by .
Multiply by .
The final solution is all the values that make true.
Substitute for into then solve for .
Replace with in the equation.
Simplify .
Multiply by .
Subtract from .
Substitute for into then solve for .
Replace with in the equation.
Simplify .
Multiply by .
Subtract from .
The solution to the system is the complete set of ordered pairs that are valid solutions.
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.
Simplify each term.
Apply the distributive property.
Multiply by .
Multiply by .
Combine the opposite terms in .
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.
Substitute and simplify.
Evaluate at and at .
Evaluate at and at .
Simplify.
Raise to the power of .
Raise to the power of .
Move the negative in front of the fraction.
Multiply by .
Multiply by .
Combine the numerators over the common denominator.
Multiply by .
Multiply by .