Find the area bounded by the curves y=x3 and y=4x.
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Step-by-Step Solution
Step 1: Find the points of intersection
To find the points where the two curves intersect, we set their equations equal to each other. This gives us an equation in terms of x$. We then factor the equation to find the values of x where the curves meet.
Step 2: Determine the intervals and upper/lower curves
From the factored equation x(x-2)(x+2) = 0$, we find the x-coordinates of the intersection points. These points divide the area into intervals. We need to determine which curve is above the other in each interval.
Step 3: Set up the integral for the area
For the interval [-2, 0] we test a point, say x =−1. y=4(−1)=−4 and y=(−1)3=−1. Since -1 > -4, y=x3 is the upper curve. For the interval [0, 2] ,wetestx=1. y=4(1)=4 and y=(1)3=1. Since 4 > 1, y=4x is the upper curve. Wait, I made a mistake in the explanation. Let's re-evaluate. For x=−1 ,y=4x=−4andy=x3=−1. So x3 is above 4x .Forx=1, y=4x=4 and y=x3=1 .So4xisabovex3. The problem statement is symmetric, so the area from -2 to 0 where x3 is above 4x will be equal to the area from 0 to 2 where 4x is above x3. Therefore, the total area is the sum of the absolute differences of the integrals.
Step 4: Corrected integral setup
Based on the analysis of the upper and lower curves, the integral for the area from x=−2 to x=0 is ∫−20(x3−4x)dx and the integral for the area from x =0tox=2is∫02(4x−x3)dx. We sum these two integrals to get the total bounded area.
Step 5: Evaluate the integrals
We find the antiderivative for each integrand. Then, we evaluate the definite integrals using the Fundamental Theorem of Calculus, substituting the limits of integration.
Step 6: Calculate the total area
We substitute the limits of integration into the antiderivatives and perform the arithmetic. The first integral evaluates to 0 - (4 - 8) = 4 The second integral evaluates to (8−4)−0=4. Summing these values gives the total bounded area.