Find the Area Between the Curves y=2x+24 , y=x^2

Problem

y=2*x+24,y=x2

Solution

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

x2=2*x+24

2. Solve the quadratic equation by moving all terms to one side and factoring.

x2−2*x−24=0

(x−6)*(x+4)=0

x=−4,x=6

3. Determine the upper function by testing a value between the intersections, such as x=0 Since 2(0) + 24 = 24a*n*d^2 = 0,t*h*e*l*i*n*e = 2x + 24$ is the upper function.

4. Set up the definite integral for the area by subtracting the lower function from the upper function over the interval [−4,6]

A=(∫_−4^6)((2*x+24−x2)*d(x))

5. Find the antiderivative of the integrand.

(∫_^)((2*x+24−x2)*d(x))=x2+24*x−(x3)/3

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

A=(6+24*(6)−6/3)−((−4)2+24*(−4)−((−4)3)/3)

A=(36+144−72)−(16−96+64/3)

A=108−(−80+64/3)

A=108−(−240/3+64/3)

A=108−(−176/3)

A=324/3+176/3

A=500/3

Final Answer

(∫_−4^6)((2*x+24−x2)*d(x))=500/3

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