The area enclosed within the ellipse `4x^(2)+9y^(2)=36` is

Options

A. `4pi`
B. `6pi`
C. `9pi`
D. `36pi`

Answer

To find the area enclosed within the ellipse given by the equation \(4x^2 + 9y^2 = 36\), we can follow these steps:

### Step 1: Rewrite the equation of the ellipse in standard form.
We start with the equation:
\[ 4x^2 + 9y^2 = 36 \]
To convert it into standard form, we divide the entire equation by 36:
\[ \frac{4x^2}{36} + \frac{9y^2}{36} = 1 \]
This simplifies to:
\[ \frac{x^2}{9} + \frac{y^2}{4} = 1 \]
Now we can identify \(a^2\) and \(b^2\):
\[ a^2 = 9 \quad \Rightarrow \quad a = 3 \]
\[ b^2 = 4 \quad \Rightarrow \quad b = 2 \]

### Step 2: Use the formula for the area of an ellipse.
The area \(A\) of an ellipse is given by the formula:
\[ A = \pi \cdot a \cdot b \]
Substituting the values of \(a\) and \(b\):
\[ A = \pi \cdot 3 \cdot 2 \]

### Step 3: Calculate the area.
Now we calculate the area:
\[ A = 6\pi \]

### Conclusion
The area enclosed within the ellipse is:
\[ \boxed{6\pi} \text{ square units} \]

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