Find the area of the region enclosed by the graphs of
( )
3 2 f x x x = −
and
g x x ( ) = 2

To find the area of the region enclosed by the graphs of the functions:

\[
f(x) = x^3 - x^2
\]
and
\[
g(x) = 2
\]

we follow these steps:

### 1. **Find the Points of Intersection:**

We set \( f(x) = g(x) \) to find where the two graphs intersect.

\[
x^3 - x^2 = 2
\]

Rearranging:

\[
x^3 - x^2 - 2 = 0
\]

To solve this cubic equation, we attempt trial values. Testing \( x = 2 \):

\[
2^3 - 2^2 - 2 = 8 - 4 - 2 = 2
\]

So, \( x = 2 \) is a root. Now, we divide \( x^3 - x^2 - 2 \) by \( (x - 2) \) to get the remaining factors.

Performing synthetic division or polynomial division, we get:

\[
x^3 - x^2 - 2 = (x - 2)(x^2 + x + 1)
\]

The quadratic \( x^2 + x + 1 = 0 \) does not have real roots (its discriminant is negative), so the only real intersection is at \( x = 2 \).

Now, we check the behavior of the functions for other points. The region of interest lies between \( x = 0 \) and \( x = 2 \), since at \( x = 0 \):

\[
f(0) = 0^3 - 0^2 = 0 \quad \text{and} \quad g(0) = 2
\]

Thus, the two curves enclose an area between \( x = 0 \) and \( x = 2 \).

### 2. **Set Up the Integral:**

The area between two curves \( y = f(x) \) and \( y = g(x) \) from \( x = a \) to \( x = b \) is given by:

\[
\text{Area} = \int_{a}^{b} \left( g(x) - f(x) \right) dx
\]

In this case, \( g(x) = 2 \) and \( f(x) = x^3 - x^2 \), so:

\[
\text{Area} = \int_{0}^{2} \left( 2 - (x^3 - x^2) \right) dx
\]

This simplifies to:

\[
\text{Area} = \int_{0}^{2} \left( 2 - x^3 + x^2 \right) dx
\]

### 3. **Compute the Integral:**

Now, we compute the integral:

\[
\int_{0}^{2} (2 - x^3 + x^2) dx = \left[ 2x - \frac{x^4}{4} + \frac{x^3}{3} \right]_{0}^{2}
\]

Evaluating at \( x = 2 \):

\[
2(2) - \frac{2^4}{4} + \frac{2^3}{3} = 4 - \frac{16}{4} + \frac{8}{3} = 4 - 4 + \frac{8}{3} = \frac{8}{3}
\]

Evaluating at \( x = 0 \) results in 0.

### 4. **Final Answer:**

Thus, the area of the region enclosed by the curves is:

\[
\boxed{\frac{8}{3}}
\]

Would you like more details or have any questions?

Here are 5 related questions:
1. How do we solve cubic equations with real and complex roots?
2. What are the methods to approximate the roots of cubic equations?
3. How do we calculate the area between two curves with multiple intersections?
4. How does synthetic division work when dividing polynomials?
5. How do we handle cases where the integral produces negative values?

**Tip:** Always check the points of intersection carefully when solving area problems between curves, as these determine your limits of integration.

Find the area of the region enclosed by the graphs of f(x) = x^3 - x^2 and g(x) = 2

Learn how to calculate the area between the curves f(x) = x^3 - x^2 and g(x) = 2, including finding points of intersection and setting up the integral. This detailed solution covers cubic functions, synthetic division, and applying the Fundamental Theorem of Calculus. Suitable for students in Grades 11-12 or college-level calculus.

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