Find the Area Between the Curves y=x+12 , y=x^2

Problem

y=x+12,y=x2

Solution

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

x2=x+12

x2−x−12=0

(x−4)*(x+3)=0

x=−3,x=4

2. Identify the upper and lower functions on the interval [−3,4] by testing a point such as x=0 Since 0+12>0 the line y=x+12 is the upper function.

3. Set up the definite integral for the area by subtracting the lower function from the upper function over the interval.

(∫_−3^4)((x+12−x2)*d(x))

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

[(x2)/2+12*x−(x3)/3]4(−3)

5. Evaluate the definite integral at the upper limit x=4

4/2+12*(4)−4/3=8+48−64/3

56−64/3=168/3−64/3=104/3

6. Evaluate the definite integral at the lower limit x=−3

((−3)2)/2+12*(−3)−((−3)3)/3=9/2−36+9

9/2−27=9/2−54/2=−45/2

7. Subtract the lower limit value from the upper limit value to find the total area.

104/3−(−45/2)=104/3+45/2

208/6+135/6=343/6

Final Answer

(∫_−3^4)((x+12−x2)*d(x))=343/6

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