Find the Area Between the Curves y=x^3 , y=4x

Problem

y=x3,y=4*x

Solution

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

x3=4*x

x3−4*x=0

x*(x2−4)=0

x*(x−2)*(x+2)=0

x=−2,0,2

2. Determine the relative positions of the curves on the intervals [−2,0] and [0,2] For [−2,0] x3≥4*x For [0,2] 4*x≥x3

3. Set up the definite integrals for the area. Due to the symmetry of the functions (both are odd), the area from [−2,0] is equal to the area from [0,2]

Area=(∫_−2^0)((x3−4*x)*d(x))+(∫_0^2)((4*x−x3)*d(x))

4. Evaluate the first integral using the power rule.

(∫_−2^0)((x3−4*x)*d(x))=[(x4)/4−2*x2]0(−2)

(0)−(((−2)4)/4−2*(−2)2)

−(4−8)=4

5. Evaluate the second integral using the power rule.

(∫_0^2)((4*x−x3)*d(x))=[2*x2−(x4)/4]20

(2*(2)2−2/4)−(0)

8−4=4

6. Sum the areas to find the total area between the curves.

Total Area=4+4=8

Final Answer

Area=(∫_−2^0)((x3−4*x)*d(x))+(∫_0^2)((4*x−x3)*d(x))=8

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