Find the Area Between the Curves 2x+y^2=8 , x=y

Problem

2*x+y2=8,x=y

Solution

1. Express the equations in terms of y to set up the integral with respect to the yaxis.

x=4−(y2)/2

x=y

2. Find the intersection points by setting the two expressions for x equal to each other.

4−(y2)/2=y

8−y2=2*y

y2+2*y−8=0

(y+4)*(y−2)=0

y=−4,y=2

3. Determine the integrand by identifying which curve is further to the right on the interval [−4,2]

Test *y=0⇒4−0/2=4,x=0

4>0⇒4−(y2)/2≥y

4. Set up the definite integral for the area A between the curves from y=−4 to y=2

A=(∫_−4^2)((4−(y2)/2−y)*d(y))

5. Integrate the expression with respect to y

(∫_^)((4−(y2)/2−y)*d(y))=4*y−(y3)/6−(y2)/2

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

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

A=(8−4/3−2)−(−16+32/3−8)

A=14/3−(−40/3)

A=54/3=18

Final Answer

(∫_−4^2)((4−(y2)/2−y)*d(y))=18

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