The volume of the solid obtained by revolving about y-axis the area enclosed between the ellipse `x^(2)+9y^(2)=9` and the strangth line `x+3y=3`, in the first quadrant is

Options

A. `3pi`
B. `4pi`
C. `6pi`
D. `9pi`

Answer

To find the volume of the solid obtained by revolving the area enclosed between the ellipse \(x^2 + 9y^2 = 9\) and the straight line \(x + 3y = 3\) about the y-axis in the first quadrant, we can follow these steps:

### Step 1: Rewrite the equations in standard form
The equation of the ellipse can be rewritten as:
\[ \frac{x^2}{9} + \frac{y^2}{1} = 1 \]
This shows that the semi-major axis is 3 (along the x-axis) and the semi-minor axis is 1 (along the y-axis).

The equation of the line can be rewritten as:
\[ x = 3 - 3y \]

### Step 2: Find the points of intersection
To find the points of intersection between the ellipse and the line, substitute \(x = 3 - 3y\) into the ellipse equation:
\[ (3 - 3y)^2 + 9y^2 = 9 \]
Expanding and simplifying:
\[ 9 - 18y + 9y^2 + 9y^2 = 9 \]
\[ 18y^2 - 18y = 0 \]
Factoring gives:
\[ 18y(y - 1) = 0 \]
Thus, \(y = 0\) or \(y = 1\).

Substituting these values back to find \(x\):
- For \(y = 0\): \(x = 3 - 3(0) = 3\)
- For \(y = 1\): \(x = 3 - 3(1) = 0\)

The points of intersection are \((3, 0)\) and \((0, 1)\).

### Step 3: Set up the volume integral
The volume \(V\) of the solid of revolution about the y-axis can be calculated using the formula:
\[ V = \pi \int_{y_1}^{y_2} (R^2 - r^2) \, dy \]
where \(R\) is the outer radius (from the ellipse) and \(r\) is the inner radius (from the line).

From the ellipse:
\[ x^2 = 9 - 9y^2 \quad \text{(outer radius)} \]
From the line:
\[ x^2 = (3 - 3y)^2 = 9 - 18y + 9y^2 \quad \text{(inner radius)} \]

### Step 4: Determine the limits of integration
The limits of integration are from \(y = 0\) to \(y = 1\).

### Step 5: Set up the integral
Thus, the volume integral becomes:
\[ V = \pi \int_{0}^{1} \left( (9 - 9y^2) - (9 - 18y + 9y^2) \right) \, dy \]
Simplifying the integrand:
\[ V = \pi \int_{0}^{1} (9 - 9y^2 - 9 + 18y - 9y^2) \, dy = \pi \int_{0}^{1} (18y - 18y^2) \, dy \]
\[ = 18\pi \int_{0}^{1} (y - y^2) \, dy \]

### Step 6: Evaluate the integral
Calculating the integral:
\[ \int (y - y^2) \, dy = \frac{y^2}{2} - \frac{y^3}{3} \]
Evaluating from 0 to 1:
\[ = \left[ \frac{1^2}{2} - \frac{1^3}{3} \right] - \left[ \frac{0^2}{2} - \frac{0^3}{3} \right] = \frac{1}{2} - \frac{1}{3} = \frac{3}{6} - \frac{2}{6} = \frac{1}{6} \]

### Step 7: Final volume calculation
Substituting back into the volume formula:
\[ V = 18\pi \cdot \frac{1}{6} = 3\pi \]

### Conclusion
Thus, the volume of the solid obtained by revolving the area enclosed between the ellipse and the line about the y-axis in the first quadrant is:
\[ \boxed{3\pi} \]