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−2=0

(x+2)*(x−1)=0

x=−2,x=1

2. Identify the upper and lower functions on the interval [−2,1] by testing a value such as x=0

(y_1)(0)=4−0=4

(y_2)(0)=0+2=2

4>2⇒y=4−x2* is the upper curve.

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

A=(∫_−2^1)((4−x2−(x+2))*d(x))

4. Simplify the integrand before integrating.

A=(∫_−2^1)((2−x−x2)*d(x))

5. Find the antiderivative of the expression.

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

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

A=[2*(1)−((1)2)/2−((1)3)/3]−[2*(−2)−((−2)2)/2−((−2)3)/3]

A=[2−1/2−1/3]−[−4−2+8/3]

A=[12/6−3/6−2/6]−[−18/3+8/3]

A=7/6−(−10/3)

A=7/6+20/6

A=27/6

7. Simplify the final result to its lowest terms.

A=9/2

Final Answer

(∫_−2^1)((4−x2−(x+2))*d(x))=9/2

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