Find the Area Between the Curves y=7-x^2 , y=x+5

Problem

y=7−x2,y=x+5

Solution

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

7−x2=x+5

2. Rearrange the equation into a standard quadratic form to solve for x

x2+x−2=0

3. Factor the quadratic to find the specific xvalues where the curves cross.

(x+2)*(x−1)=0

x=−2,x=1

4. Determine the upper function by testing a value between the intersections, such as x=0 Since 7−0=7 and 0 + 5 = 5,t*h*e*c*u*r*v*e = 7 - x^2$ is the upper boundary.

5. Set up the integral for the area using the formula A=(∫_a^b)((upper−lower)*d(x))

A=(∫_−2^1)((7−x2−(x+5))*d(x))

6. Simplify the integrand before performing the integration.

A=(∫_−2^1)((−x2−x+2)*d(x))

7. Integrate the expression using the power rule.

A=[−(x3)/3−(x2)/2+2*x]1(−2)

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

A=(−1/3−1/2+2*(1))−(−((−2)3)/3−((−2)2)/2+2*(−2))

A=(−1/3−1/2+2)−(8/3−2−4)

A=7/6−(−10/3)

A=7/6+20/6

A=27/6

9. Simplify the final result to its lowest terms.

A=9/2

Final Answer

(∫_−2^1)((7−x2−(x+5))*d(x))=9/2

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