Question

Find the area of the region bounded by the ellipse \(\frac{x^2}{9^2} + \frac{y^2}{4^2} = 1\).

Hint:

The area formula for an ellipse, \(A = \pi ab\), is a direct generalization of the area of a circle. For a circle, the semi-major and semi-minor axes are both equal to the radius (\(a=b=r\)), and the formula becomes \(A = \pi(r)(r) = \pi r^2\). Memorizing this formula is essential for competitive exams.

Answer 1

Step 1: Understanding the Concept:
The problem asks for the total area enclosed by an ellipse given in its standard form. This can be found directly using the standard formula for the area of an ellipse.
Step 2: Key Formula or Approach:
The standard equation of an ellipse centered at the origin is \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\).
The area (A) enclosed by such an ellipse is given by the formula: \[ A = \pi ab \] where \(a\) and \(b\) are the lengths of the semi-major and semi-minor axes, respectively.
Step 3: Detailed Explanation:
The given equation of the ellipse is: \[ \frac{x^2}{9^2} + \frac{y^2}{4^2} = 1 \] By comparing this with the standard form \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), we can identify the values of \(a\) and \(b\).
\[ a^2 = 9^2 \implies a = 9 \] \[ b^2 = 4^2 \implies b = 4 \] Now, we apply the formula for the area of the ellipse: \[ A = \pi ab \] Substituting the values of \(a\) and \(b\): \[ A = \pi (9)(4) = 36\pi \] Step 4: Final Answer:
The area of the region bounded by the ellipse is \(36\pi\) square units.