Find the Area Between the Curves y=5-x^2 , y=x-1

Problem

y=5−x2,y=x−1

Solution

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

5−x2=x−1

2. Rearrange into a quadratic equation by moving all terms to one side.

x2+x−6=0

3. Factor the quadratic to solve for the xvalues.

(x+3)*(x−2)=0

x=−3,x=2

4. Identify the upper and lower functions on the interval [−3,2] By testing a point like x=0 we see 5−(0)2=5 and 0 - 1 = -1,s(o) = 5 - x^2$ is the upper curve.

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

A=(∫_−3^2)((5−x2−(x−1))*d(x))

6. Simplify the integrand before integrating.

A=(∫_−3^2)((−x2−x+6)*d(x))

7. Find the antiderivative of the expression.

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

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

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

9. Calculate the numerical values for each part.

A=(−8/3−2+12)−(9−9/2−18)

A=22/3−(−27/2)

10. Find a common denominator to sum the fractions.

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

Final Answer

(∫_−3^2)((5−x2−(x−1))*d(x))=125/6

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