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

,
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 .
Since is on the right side of the equation, switch the sides so it is on the left side of the equation.
Set the equation equal to zero.
Move to the left side of the equation by subtracting it from both sides.
Subtract from .
Factor the left side of the equation.
Factor out of .
Factor out of .
Factor out of .
Factor out of .
Replace the left side with the factored expression.
Divide each term in by .
Cancel the common factor of .
Cancel the common factor.
Divide by .
Divide by .
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 .
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 .
Simplify each term.
Multiply by .
Raising to any positive power yields .
Multiply by .
Substitute for into then solve for .
Replace with in the equation.
Simplify .
Simplify each term.
Multiply by .
Raise to the power of .
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.
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 .
Simplify.
Raise to the power of .
Cancel the common factor of and .
Factor out of .
Cancel the common factors.
Factor out of .
Cancel the common factor.
Rewrite the expression.
Divide by .
Raising to any positive power yields .
Cancel the common factor of and .
Factor out of .
Cancel the common factors.
Factor out of .
Cancel the common factor.
Rewrite the expression.
Divide by .
Multiply by .
Multiply by .
Raise to the power of .
Raising to any positive power yields .
Cancel the common factor of and .
Factor out of .
Cancel the common factors.
Factor out of .
Cancel the common factor.
Rewrite the expression.
Divide by .
Multiply by .
Combine and .
Multiply by .
Move the negative in front of the fraction.
To write as a fraction with a common denominator, multiply by .
Combine and .
Combine the numerators over the common denominator.
Simplify the numerator.
Multiply by .
Subtract from .
Find the Area Between the Curves y=x^2 , y=4x-x^2

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