3.7 Q-10

Question Statement

Find the area above the $x$-axis bounded by the curve $y^2 = 3 - x$ from $x = -1$ to $x = 2$.

---

Background and Explanation

This problem requires understanding of how to calculate the area between a curve and the x-axis. The given equation is a relation between $y$ and $x$. To find the area, we need to:

• Solve for $y$ in terms of $x$.
• Use definite integrals to find the area under the curve.

In this case, the curve $y^2 = 3 - x$ has two branches, but we are concerned with the positive branch because the area we need to find lies above the x-axis.

---

Solution

1. Start by solving for $y$:

$$y^2 = 3 - x \quad \Rightarrow \quad y = \pm \sqrt{3 - x}$$

Since we are looking for the area above the x-axis, we choose the positive branch:

$$y = \sqrt{3 - x}$$

This gives us the equation of the curve above the x-axis.

2. Determine the limits of integration: The problem specifies the range from $x = -1$ to $x = 2$. Therefore, we need to find the area under the curve from $x = -1$ to $x = 2$.

3. Set up the integral for the area: The area is found by integrating $y = \sqrt{3 - x}$ from $x = -1$ to $x = 2$:

$$\text{Area} = \int_{-1}^{2} \sqrt{3 - x} , dx$$

Since we are integrating with respect to $x$, we proceed with solving this integral.

4. Solve the integral: First, rewrite the integral:

$$\text{Area} = \int_{-1}^{2} (3 - x)^{1/2} , dx$$

To simplify, use the substitution method. The integral becomes:

$$= \int_{-1}^{2} (3 - x)^{1/2} (-1) , dx$$

Now, apply the power rule for integration:

$$\int (3 - x)^{1/2} dx = \frac{(3 - x)^{3/2}}{3/2} = \frac{2}{3} (3 - x)^{3/2}$$

5. Evaluate the definite integral: Now, substitute the limits $x = -1$ and $x = 2$:

$$\text{Area} = \frac{2}{3} \left[ (3 - (-1))^{3/2} - (3 - 2)^{3/2} \right]$$

Simplify the terms inside the parentheses:

$$= \frac{2}{3} \left[ (4)^{3/2} - (1)^{3/2} \right]$$

Calculate the values:

$$= \frac{2}{3} \left[ 8 - 1 \right]$$ $$= \frac{2}{3} \times 7 = \frac{14}{3}$$

Thus, the area is:

$$\text{Area} = \frac{14}{3} , \text{(sq. units)}$$

---

Key Formulas or Methods Used

• Power rule for integration:

$$\int (a - x)^n , dx = \frac{(a - x)^{n+1}}{n+1}$$

• Substitution method (in this case, simple integration of a binomial expression).

---

Summary of Steps

1. Solve for $y$: $y = \sqrt{3 - x}$.
2. Set up the integral: $\int_{-1}^{2} \sqrt{3 - x} , dx$.
3. Apply the power rule and simplify the integral.
4. Evaluate the integral using the limits of integration.
5. Final answer: $\frac{14}{3}$ square units.