To find the area of the region bounded by the curves $y = \sqrt{x+1}$, $y + x = 1$, and the $x$-axis, we need to determine the points of intersection and then set up the appropriate integrals.

1. Finding the points of intersection

First, we need to find the points where $y = \sqrt{x+1}$ and $y + x = 1$ intersect.

Set $y = \sqrt{x+1}$ equal to $y = 1 - x$: $\sqrt{x+1} = 1 - x$

Square both sides to remove the square root: $(\sqrt{x+1})^2 = (1 - x)^2$ $x + 1 = 1 - 2x + x^2$

Rearrange the equation to standard quadratic form: $x^2 - 3x = 0$ $x(x - 3) = 0$

So, $x = 0$ or $x = 3$.

Now, substitute these $x$-values back into $y = 1 - x$ to find the corresponding $y$-values:

• For $x = 0$: $y = 1 - 0 = 1$
• For $x = 3$: $y = 1 - 3 = -2$

Since $y = \sqrt{x+1}$ only provides non-negative $y$-values, the relevant intersection points are at $(0, 1)$ and $(3, -2)$.

2. Setting up the integrals

We will split the integral into two parts:

• From $x = 0$ to $x = 1$ (where $\sqrt{x+1}$ is above the $x$-axis)
• From $x = 1$ to $x = 3$ (where $y + x = 1$ is above the $x$-axis)

The area $A$ can be calculated as: $A = \int_{0}^{1} \sqrt{x+1} \, dx + \int_{1}^{3} (1 - x) \, dx$

3. Computing the integrals

First Integral:

$\int_{0}^{1} \sqrt{x+1} \, dx$

Let $u = x + 1$. Then, $du = dx$. When $x = 0$, $u = 1$. When $x = 1$, $u = 2$. $\int_{1}^{2} \sqrt{u} \, du = \int_{1}^{2} u^{1/2} \, du$ $= \left[ \frac{2}{3} u^{3/2} \right]_{1}^{2}$ $= \frac{2}{3} (2^{3/2} - 1^{3/2})$ $= \frac{2}{3} (2\sqrt{2} - 1)$

Second Integral:

$\int_{1}^{3} (1 - x) \, dx$ $= \left[ x - \frac{x^2}{2} \right]_{1}^{3}$ $= \left[ (3 - \frac{9}{2}) - (1 - \frac{1}{2}) \right]$ $= \left( 3 - 4.5 \right) - \left( 1 - 0.5 \right)$ $= -1.5 - 0.5$ $= -2$

4. Total Area

The total area $A$ is the sum of the areas of the two integrals: $A = \frac{2}{3} (2\sqrt{2} - 1) + (-2)$ $= \frac{4\sqrt{2} - 2}{3} - 2$ $= \frac{4\sqrt{2} - 2 - 6}{3}$ $= \frac{4\sqrt{2} - 8}{3}$

So, the area of the region is: $\boxed{\frac{4\sqrt{2} - 8}{3}}$

If you have any questions or need further details, let me know!

Here are 5 questions you might want to ask next:

1. How do you find the volume of a solid of revolution?
2. What is the method of partial fractions in calculus?
3. Can you explain the Fundamental Theorem of Calculus?
4. How do you solve a system of linear equations using matrices?
5. What are the properties of definite integrals?

Tip: When solving integrals, always consider whether you can simplify the problem using substitution or by breaking the integral into simpler parts.