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

Problem

y=4−x2,y=x−2

Solution

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

4−x2=x−2

x2+x−6=0

(x+3)*(x−2)=0

x=−3,x=2

2. Identify the upper and lower functions on the interval [−3,2] by testing a point such as x=0

(y_1)(0)=4−0=4

(y_2)(0)=0−2=−2

4>−2⇒(y_top)=4−x2,(y_bottom)=x−2

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

A=(∫_−3^2)((4−x2−(x−2))*d(x))

A=(∫_−3^2)((−x2−x+6)*d(x))

4. Integrate the expression using the power rule.

(∫_^)((−x2−x+6)*d(x))=−(x3)/3−(x2)/2+6*x

5. Evaluate the definite integral at the boundaries x=2 and x=−3

A=(−2/3−2/2+6*(2))−(−((−3)3)/3−((−3)2)/2+6*(−3))

A=(−8/3−2+12)−(9−9/2−18)

A=22/3−(−27/2)

A=44/6+81/6=125/6

Final Answer

(∫_−3^2)((4−x2−(x−2))*d(x))=125/6

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