Find the area bounded by the curves $y = x^3$ and $y = 4x$.

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, $y = x^3$ is the upper curve. For the interval [0, 2] $, we test x = 1$. $y = 4(1) = 4$ 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 $x = -1$ $, y = 4x = -4 and y = x^3 = -1$. So $x^3$ is above 4x $. For x = 1$, $y = 4x = 4$ and $y = x^3 = 1$. So 4x is above $x^3$. The problem statement is symmetric, so the area from -2 to 0 where $x^3$ is above 4x will be equal to the area from 0 to 2 where 4x is above $x^3$. 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 $\int_{-2}^{0} (x^3 - 4x) dx$, and the integral for the area from $x = 0$ to $x = 2$ is $\int_{0}^{2} (4x - x^3) dx$. 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.

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