Question

Find the area of the bounded region of

\[ \frac{x^2}{16} + \frac{y^2}{9} = 1 \]

Hint:

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

Answer 1

Step 1: Recognize that this is the equation of an ellipse in standard form: \[ \frac{x^2}{16} + \frac{y^2}{9} = 1 \] where \( a = 4 \) and \( b = 3 \) are the semi-major and semi-minor axes, respectively. 

Step 2: The area of an ellipse is given by the formula: \[ A = \pi \cdot a \cdot b \] Substitute the values of \( a \) and \( b \): \[ A = \pi \cdot 4 \cdot 3 = 12\pi. \] Thus, the area of the bounded region is \( 12\pi \).