Question

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

Hint:

The area of an ellipse is given by \( A = \pi \cdot a \cdot b \), where \( a \) and \( b \) are the lengths of the semi-major and semi-minor axes.

Answer 1

The equation of the ellipse is given by: \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1. \] To find the area of the region bounded by the ellipse, we use the formula for the area of an ellipse: \[ A = \pi \cdot a \cdot b. \] This is the standard result for the area of an ellipse, where \( a \) and \( b \) are the semi-major and semi-minor axes, respectively. Conclusion: The area of the region bounded by the ellipse is: \[ \boxed{A = \pi \cdot a \cdot b}. \]