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

Problem

y=2*x−x2,y=2*x−4

Solution

1. Find the intersection points by setting the two equations equal to each other to determine the limits of integration.

2*x−x2=2*x−4

−x2=−4

x2=4

x=±2

2. Determine the upper and lower functions on the interval [−2,2] by testing a point such as x=0

(y_1)(0)=2*(0)−(0)2=0

(y_2)(0)=2*(0)−4=−4

Since 0>−4 the upper function is ƒ(x)=2*x−x2 and the lower function is g(x)=2*x−4

3. Set up the integral for the area A using the formula A=(∫_a^b)((ƒ(x)−g(x))*d(x))

A=(∫_−2^2)((2*x−x2−(2*x−4))*d(x))

4. Simplify the integrand before integrating.

A=(∫_−2^2)((4−x2)*d(x))

5. Integrate the expression using the power rule.

A=[4*x−(x3)/3]2(−2)

6. Evaluate the definite integral at the upper and lower limits.

A=(4*(2)−2/3)−(4*(−2)−((−2)3)/3)

A=(8−8/3)−(−8+8/3)

A=16/3−(−16/3)

A=32/3

Final Answer

(∫_−2^2)((2*x−x2−(2*x−4))*d(x))=32/3

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