Key Concepts and Formulas

• Area Between Curves (Integration with respect to y): If a region is bounded on the right by $x = f(y)$ and on the left by $x = g(y)$ from $y=c$ to $y=d$, where $f(y) \ge g(y)$ for all $y \in [c,d]$, the area is given by: $A = \int_c^d [f(y) - g(y)] \, dy$
• Definite Integral: The definite integral of a function $f(x)$ from $a$ to $b$, denoted by $\int_a^b f(x) \, dx$, represents the area under the curve $y = f(x)$ between the limits $x = a$ and $x = b$.
• Fundamental Theorem of Calculus: If $F(x)$ is an antiderivative of $f(x)$, then $\int_a^b f(x) \, dx = F(b) - F(a)$.

Step-by-Step Solution

Step 1: Understanding and Visualizing the Curves

We need to find the area bounded by $y = \sqrt{x}$, $2y - x + 3 = 0$, the x-axis ($y=0$), and lying in the first quadrant. Visualizing these curves is crucial to determine the region and choose an appropriate integration strategy.

• $y = \sqrt{x}$ is the upper half of a parabola opening to the right.
• $2y - x + 3 = 0$ is a line.
• $x$-axis is $y = 0$.
• First quadrant restricts us to $x \ge 0$ and $y \ge 0$.

Step 2: Finding Intersection Points

We need to find the intersection points of the curves to define the limits of integration.

1. Intersection of $y = \sqrt{x}$ and $2y - x + 3 = 0$:

   • Substitute $x = y^2$ (from $y = \sqrt{x}$) into the line equation: $2y - y^2 + 3 = 0$.
   • Rearrange: $y^2 - 2y - 3 = 0$.
   • Factor: $(y - 3)(y + 1) = 0$.
   • Solutions: $y = 3$ or $y = -1$. Since $y = \sqrt{x}$, we must have $y \ge 0$, so we discard $y = -1$.
   • If $y = 3$, then $x = y^2 = 3^2 = 9$. The intersection point is $(9, 3)$.

2. Intersection of $y = \sqrt{x}$ and $y = 0$ (x-axis):

   • Substitute $y = 0$ into $y = \sqrt{x}$: $0 = \sqrt{x}$, so $x = 0$. The intersection point is $(0, 0)$.

3. Intersection of $2y - x + 3 = 0$ and $y = 0$ (x-axis):

   • Substitute $y = 0$ into $2y - x + 3 = 0$: $2(0) - x + 3 = 0$, so $x = 3$. The intersection point is $(3, 0)$.

Step 3: Choosing the Integration Variable

We can integrate with respect to $x$ or $y$. Let's analyze both:

• Integrating with respect to $x$: We'd need to split the integral into two parts: from $x = 0$ to $x = 3$ and from $x = 3$ to $x = 9$, because the lower bounding curve changes at $x=3$.
• Integrating with respect to $y$: We can express both curves as $x = f(y)$: $x = y^2$ and $x = 2y + 3$. The limits of integration will be from $y = 0$ to $y = 3$. The line is always to the right of the parabola in this region.

Integrating with respect to $y$ requires only one integral and is therefore simpler.

Step 4: Setting up the Definite Integral

The area is given by: $A = \int_0^3 [(2y + 3) - y^2] \, dy$ The limits of integration are $0$ and $3$ because those are the $y$-values where the region starts and ends. The function being integrated is (right curve) - (left curve) = (line) - (parabola).

Step 5: Evaluating the Definite Integral

$A = \int_0^3 (2y + 3 - y^2) \, dy = \left[ y^2 + 3y - \frac{y^3}{3} \right]_0^3$ $A = \left( (3)^2 + 3(3) - \frac{(3)^3}{3} \right) - \left( (0)^2 + 3(0) - \frac{(0)^3}{3} \right)$ $A = (9 + 9 - 9) - (0) = 9$

Common Mistakes & Tips

• Sketching is Essential: Always sketch the region to visualize the problem and identify the correct limits of integration.
• Choose the Easier Integration Variable: Carefully consider whether integrating with respect to $x$ or $y$ will result in a simpler integral.
• Correctly Identify Right/Left or Above/Below: Make sure you are subtracting the correct functions when setting up the integral.

Summary

By visualizing the region bounded by the given curves, finding the intersection points, and strategically choosing to integrate with respect to $y$, we simplified the problem to a single definite integral. Evaluating this integral, we found the area to be 9 square units.

Final Answer

The final answer is \boxed{9}, which corresponds to option (A).