Question

Find the area of the region enclosed by the ellipse given parametrically by the coordinates \( x = 2\sin\theta \) and \( y = 3\cos\theta \) where \( 0 \le \theta \le 2\pi \).

• A. \( 6\pi \)
• B. \( 12\pi \)
• C. \( 5\pi \)
• D. \( 13\pi \)

Hint:

For parametric coordinate boundaries that fit the form \( x = a\sin\theta \) and \( y = b\cos\theta \), the coefficients are always your semi-axes values. You can skip the Cartesian conversion entirely and solve directly using \( \text{Area} = \pi a b \).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
An ellipse is a locus of points in a plane such that the sum of the distances from two fixed points (foci) is constant.
In Cartesian geometry, a standard ellipse centered at the origin is represented by the equation \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \).
The area enclosed by such an ellipse is given by the formula \( \text{Area} = \pi ab \), where \( a \) and \( b \) are the lengths of the semi-major and semi-minor axes.
Parametric equations define the coordinates of the points on the curve as functions of a third variable, called a parameter (often denoted as \( \theta \) or \( t \)).
By converting parametric equations back to Cartesian form, we can easily identify the axes lengths and compute the area.
Step 2: Key Formula or Approach:
Method 1: Identify the semi-axes \( a \) and \( b \) from the parametric coefficients and apply \( A = \pi ab \).
Method 2: Convert to Cartesian form using the identity \( \sin^2 \theta + \cos^2 \theta = 1 \).
Method 3: Use the parametric integral area formula: \( A = \int_{0}^{2\pi} x \frac{dy}{d\theta} d\theta \).
Step 3: Detailed Explanation:
We are given the parametric coordinates of the curve:
\( x = 2 \sin \theta \)
\( y = 3 \cos \theta \)
First, we isolate the trigonometric terms:
\( \sin \theta = \frac{x}{2} \)
\( \cos \theta = \frac{y}{3} \)
Using the fundamental trigonometric identity \( \sin^2 \theta + \cos^2 \theta = 1 \), we substitute our expressions:
\[ \left( \frac{x}{2} \right)^2 + \left( \frac{y}{3} \right)^2 = 1 \]
Simplifying the squares:
\[ \frac{x^2}{4} + \frac{y^2}{9} = 1 \]
Comparing this to the standard Cartesian form \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), we can extract the values for \( a \) and \( b \):
\( a^2 = 4 \implies a = 2 \)
\( b^2 = 9 \implies b = 3 \)
Now, we apply the area formula for an ellipse:
\[ \text{Area} = \pi \cdot a \cdot b \]
Substituting our identified values:
\[ \text{Area} = \pi \cdot 2 \cdot 3 \]
\[ \text{Area} = 6\pi \]
This calculation represents the total space enclosed within the boundary defined by the parametric equations over the full rotation from \( 0 \) to \( 2\pi \).
Step 4: Final Answer:
The area of the region enclosed by the ellipse is \( 6\pi \).
This is provided in Option (A).