Find the area of the region that is bounded above by the curve  
f
(
x
)
=
(
x
+
2
)
2
  and the line  
g
(
x
)
=
4
−
x
  and bounded below by the  
x
 -axis.



To find the area of the region bounded by the curve \( f(x) = (x + 2)^2 \), the line \( g(x) = 4 - x \), and the \( x \)-axis, we will break the problem into three steps:

1. **Find the points of intersection** between the curve \( f(x) \) and the line \( g(x) \).
2. **Determine the definite integrals** to find the area between these curves and the \( x \)-axis.
3. **Add the areas** from the different regions to get the total area.

### Step 1: Find the Points of Intersection

We first find where the curve \( f(x) = (x + 2)^2 \) intersects with the line \( g(x) = 4 - x \). Set the two equations equal to each other:

\[
(x + 2)^2 = 4 - x
\]

Expanding and simplifying:

\[
x^2 + 4x + 4 = 4 - x
\]

\[
x^2 + 5x + 0 = 0
\]

\[
x(x + 5) = 0
\]

So, \( x = 0 \) or \( x = -5 \).

The points of intersection are at \( x = 0 \) and \( x = -5 \).

### Step 2: Determine the Definite Integrals

The area under the curve \( f(x) \) from \( x = -5 \) to \( x = 0 \) and above the \( x \)-axis is given by the difference between the line \( g(x) \) and the curve \( f(x) \).

The area is:

\[
\text{Area} = \int_{-5}^{0} \left[ g(x) - f(x) \right] \, dx = \int_{-5}^{0} \left[ (4 - x) - (x + 2)^2 \right] \, dx
\]

Expanding \( (x + 2)^2 \):

\[
(x + 2)^2 = x^2 + 4x + 4
\]

So the integrand becomes:

\[
\text{Area} = \int_{-5}^{0} \left[ 4 - x - (x^2 + 4x + 4) \right] \, dx = \int_{-5}^{0} \left[ -x^2 - 5x \right] \, dx
\]

### Step 3: Compute the Integral

Compute the integral:

\[
\text{Area} = \int_{-5}^{0} \left[ -x^2 - 5x \right] \, dx
\]

Integrating term by term:

\[
\text{Area} = \left[ -\frac{x^3}{3} - \frac{5x^2}{2} \right]_{-5}^{0}
\]

Evaluate at the bounds:

\[
\text{Area} = \left( -\frac{(0)^3}{3} - \frac{5(0)^2}{2} \right) - \left( -\frac{(-5)^3}{3} - \frac{5(-5)^2}{2} \right)
\]

Simplify:

\[
\text{Area} = 0 - \left( -\frac{-125}{3} - \frac{5 \times 25}{2} \right)
\]

\[
\text{Area} = \frac{125}{3} + \frac{125}{2}
\]

To simplify:

\[
\text{Area} = \frac{250}{6} + \frac{375}{6} = \frac{625}{6}
\]

### Final Answer

The area of the region is \( \boxed{\frac{625}{6}} \).

---

Do you want more details or have any questions?

Here are 5 related questions:

1. How do you find the intersection points between a curve and a line?
2. What are definite integrals used for in geometry?
3. Can you explain how to solve a quadratic equation?
4. Why do we subtract functions to find the area between curves?
5. What is the significance of the x-axis in determining bounded areas?

**Tip:** When solving for areas between curves, always ensure the limits of integration correspond to the points where the curves intersect.

Learn how to calculate the area bounded by the curve f(x) = (x + 2)^2, the line g(x) = 4 - x, and the x-axis using definite integrals. Detailed step-by-step solution provided.

Math Problem Statement

Find the area of the region that is bounded above by the curve
f ( x )

( x + 2 ) 2 and the line
g ( x )

Solution

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