Find the Area Between the Curves y=x^2 , y=4

Problem

y=x2,y=4

Solution

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

x2=4

x=±2

2. Identify the upper and lower functions on the interval [−2,2] Since 0<4 the line y=4 is the upper function and y=x2 is the lower function.

ƒ(x)=4

g(x)=x2

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

A=(∫_−2^2)((4−x2)*d(x))

4. Find the antiderivative of the integrand.

(∫_^)((4−x2)*d(x))=4*x−(x3)/3

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

A=[4*(2)−2/3]−[4*(−2)−((−2)3)/3]

A=[8−8/3]−[−8+8/3]

A=16/3−(−16/3)

A=32/3

Final Answer

(∫_−2^2)((4−x2)*d(x))=32/3

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