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To find the area between the two curves \( f(x) = x^3 \) and \( g(x) = x \), we need to calculate the definite integral of the difference between the two functions over the interval where they intersect.

### Step 1: Find the points of intersection
To determine the limits of integration, we find the points where \( f(x) = g(x) \), i.e., where:
\[
x^3 = x
\]

Solving this equation:
\[
x^3 - x = 0
\]
\[
x(x^2 - 1) = 0
\]
\[
x(x - 1)(x + 1) = 0
\]

The solutions are \( x = 0 \), \( x = 1 \), and \( x = -1 \).

### Step 2: Set up the integral
We will compute the area between the curves from \( x = -1 \) to \( x = 1 \). Since \( g(x) = x \) is greater than \( f(x) = x^3 \) over this interval, the area \( A \) is given by:
\[
A = \int_{-1}^{1} (g(x) - f(x)) \, dx = \int_{-1}^{1} (x - x^3) \, dx
\]

### Step 3: Compute the integral
We will now compute this integral:
\[
\int_{-1}^{1} (x - x^3) \, dx = \int_{-1}^{1} x \, dx - \int_{-1}^{1} x^3 \, dx
\]

Both integrals are symmetric around zero, so:
\[
\int_{-1}^{1} x \, dx = 0
\]
\[
\int_{-1}^{1} x^3 \, dx = 0
\]

Thus, we only need to evaluate:
\[
A = 2 \times \int_{0}^{1} (x - x^3) \, dx
\]

### Step 4: Final evaluation
Now we evaluate the integral:
\[
A = 2 \times \left( \frac{x^2}{2} - \frac{x^4}{4} \right) \Bigg|_0^1
\]
\[
A = 2 \times \left( \frac{1^2}{2} - \frac{1^4}{4} \right)
\]
\[
A = 2 \times \left( \frac{1}{2} - \frac{1}{4} \right) = 2 \times \frac{1}{4} = \frac{1}{2}
\]

Thus, the area between the curves is \( \frac{1}{2} \).

The correct answer is:
\[
\boxed{c. \ \frac{1}{2}}
\]

---

Would you like further details or explanations?

Here are five related questions for further practice:
1. How do you calculate the area between two curves when one is above the other?
2. What happens if the functions intersect more than twice within the limits of integration?
3. How would the process change if the interval was from \(x = 0\) to \(x = 2\)?
4. How can we determine which function is above the other in any given interval?
5. What does the symmetry of a function imply when calculating areas?

**Tip:** Always visualize the functions to identify which one is above the other to correctly set up the integral.

¿Cuál es la medida del área que se genera entre las funciones f(x) = x^3 y g(x) = x?

Learn how to find the area between the curves f(x) = x^3 and g(x) = x using definite integrals. This calculus problem explains key concepts such as finding points of intersection, setting up the integral, and evaluating it step by step. Suitable for students in Grades 10-12.

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