Question

Find the area of the region enclosed by the ellipse \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1. \]

Hint:

To find the area of an ellipse, use the formula \( A = \pi a b \), where \( a \) and \( b \) are the lengths of the semi-major and semi-minor axes.

Answer 1

Step 1: The area of an ellipse with the equation \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) is given by the formula: \[ A = \pi a b. \] 

Step 2: Here, \( a \) and \( b \) represent the lengths of the semi-major and semi-minor axes of the ellipse, respectively. 

Step 3: Therefore, the area enclosed by the ellipse is simply: \[ A = \pi a b. \] Thus, the area of the region enclosed by the ellipse is \( \pi a b \).