Find the area enslosed by the curve:
`x =3 cost, y =2 sin t.`

Answer

To find the area enclosed by the curve given in parametric form as \( x = 3 \cos t \) and \( y = 2 \sin t \), we can follow these steps:

### Step 1: Identify the shape of the curve
The equations \( x = 3 \cos t \) and \( y = 2 \sin t \) represent an ellipse in parametric form. To confirm this, we can eliminate the parameter \( t \).

### Step 2: Eliminate the parameter
Using the identity \( \cos^2 t + \sin^2 t = 1 \):
\[ \left(\frac{x}{3}\right)^2 + \left(\frac{y}{2}\right)^2 = 1 \]
This simplifies to:
\[ \frac{x^2}{9} + \frac{y^2}{4} = 1 \]
This is the standard form of an ellipse centered at the origin with semi-major axis \( a = 3 \) and semi-minor axis \( b = 2 \).

### Step 3: Calculate the area of the ellipse
The area \( A \) of an ellipse is given by the formula:
\[ A = \pi \times a \times b \]
Substituting the values of \( a \) and \( b \):
\[ A = \pi \times 3 \times 2 = 6\pi \]

### Step 4: Conclusion
Thus, the area enclosed by the curve is:
\[ \text{Area} = 6\pi \text{ square units} \]