Question

Find the area bounded by the ellipse \(\frac{x^2}{16} + \frac{y^2}{25} = 1\).

Hint:

Memorize the formula for the area of an ellipse, Area = \(\pi ab\). Be careful to use the values of \(a\) and \(b\) (the semi-axes lengths), not \(a^2\) and \(b^2\). For a circle, \(a=b=r\), and the formula correctly simplifies to Area = \(\pi r^2\).

Answer 1

Step 1: Understanding the Concept:
The problem is to find the total area enclosed by an ellipse. The equation of an ellipse is given in standard form, from which we can identify the lengths of the semi-major and semi-minor axes.
Step 2: Key Formula or Approach:
The standard equation of an ellipse centered at the origin is: \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \] where \(a\) is the length of the semi-horizontal axis and \(b\) is the length of the semi-vertical axis. The area of such an ellipse is given by the formula: \[ \text{Area} = \pi ab \] Step 3: Detailed Explanation or Calculation:
The given equation of the ellipse is: \[ \frac{x^2}{16} + \frac{y^2}{25} = 1 \] Comparing this to the standard form \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), we can identify \(a^2\) and \(b^2\): \[ a^2 = 16 \implies a = \sqrt{16} = 4 \] \[ b^2 = 25 \implies b = \sqrt{25} = 5 \] Now, we use the formula for the area of an ellipse: \[ \text{Area} = \pi ab \] Substitute the values of \(a\) and \(b\): \[ \text{Area} = \pi (4)(5) \] \[ \text{Area} = 20\pi \] Step 4: Final Answer:
The area bounded by the given ellipse is \(20\pi\) square units.