Find the Area Between the Curves y=7-x^2 , y=x+1

Problem

y=7−x^2,y=x+1

Solution

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

7−x^2=x+1

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

x^2+x−6=0

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

(x+3)*(x−2)=0

x=−3,x=2

4. Identify the upper curve on the interval [−3,2] by testing a point like x=0 Since 7−0^2=7 and $0 + 1 = 1,t*h*e*p*a*r*a*b*o*l*a = 7 - x^2$ is the upper curve.

5. Set up the integral for the area by subtracting the lower curve from the upper curve over the interval.

A=(∫_−3^2)((7−x^2)−(x+1)*d(x))

6. Simplify the integrand before performing the integration.

A=(∫_−3^2)(−x^2−x+6*d(x))

7. Integrate the expression using the power rule for each term.

A=[−(x^3)/3−(x^2)/2+6*x]^2_(−3)

8. Evaluate the definite integral by substituting the upper limit and subtracting the value at the lower limit.

A=(−8/3−4/2+12)−(−(−27)/3−9/2−18)

A=(−8/3+10)−(9−9/2−18)

A=22/3−(−27/2)

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

Final Answer

(∫_−3^2)((7−x^2)−(x+1)*d(x))=125/6

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