Find the Area Between the Curves y=1/2x^2 , y=-x^2+6

Problem

y=1/2*x2,y=−x2+6

Solution

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

1/2*x2=−x2+6

3/2*x2=6

x2=4

x=±2

2. Identify the upper and lower functions on the interval [−2,2] by testing a point such as x=0

(y_top)=−x2+6

(y_bottom)=1/2*x2

3. Set up the integral for the area using the formula A=(∫_a^b)(((y_top)−(y_bottom))*d(x))

A=(∫_−2^2)((−x2+6−1/2*x2)*d(x))

4. Simplify the integrand by combining like terms.

A=(∫_−2^2)((−3/2*x2+6)*d(x))

5. Integrate the expression using the power rule.

A=[−1/2*x3+6*x]2(−2)

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

A=(−1/2*(2)3+6*(2))−(−1/2*(−2)3+6*(−2))

A=(−4+12)−(4−12)

A=8−(−8)

A=16

Final Answer

(∫_−2^2)((−x2+6−1/2*x2)*d(x))=16

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