Chapter 8.4, Problem 2P

Solution

To explain : The method to obtain closer and closer approximation of the ellipse's area.

To obtain closer and closer approximation of the ellipse's area the rectangle’s width should be the minimum.

Given information :

The equation of the ellipse is $\frac{x^2}{{200}^2}+\frac{y^2}{{100}^2}=1$ and the estimated area of the elliptical region by using the rectangles of width $40$ is $68800$ square meters. The estimated area of the elliptical region using the rectangles of width $20$ is $58,088\text{ square meters}$ . The area of the ellipse using formula is $62800\text{ square meters}$ .

Formula used:

Area $\left(A\right)$ of an ellipse is given by,

  $A=\pi ab$

Explanation :

The area of the ellipse using formula is $62800\text{ square meters}$ .

The estimated area of the elliptical region using the rectangles of width $20$ is $58,088\text{ square meters}$ .

The estimated area of the elliptical region by using the rectangles of width $40$ is $68800$ square meters.

Therefore, as the width of the rectangles will decrease, the estimated area will be closer to the actual area.