Chapter 6.1, Problem 33E

To determine

To determine the area of the ellipse.

Answer to Problem 33E

Area of the Ellipse is $A$$=\pi ab\text{ sq}\text{. units}\text{.}$

Explanation of Solution

Given: Ellipse

Calculation:

  

Area of the Ellipse is $A$ can be obtained as:

Area = A = 4

  $\begin{array}{l}Are=A=4\ast {\displaystyle \underset{0}{\overset{a}{\int }}ydx}=4x{\displaystyle \underset{0}{\overset{a}{\int }}\frac{b}{a}\sqrt{a^2-x^2}dx}\\ \\ \text{ =4}\ast \frac{b}{a}{\left[\frac{x}{2}\sqrt{a^2-x^2}+\frac{a^2}{2}{\sin }^{-1}(x/a)\right]}_0^a\\ \text{ = 4b/a}\left\{\left[\frac{a}{2}\sqrt{a^2-a^2}+\frac{a^2}{2}{\sin }^{-1}(a/a)\right]-\left[0+\frac{a^2}{2}{\sin }^{-1}(0)\right]\right\}\\ \text{ = 4b/a}\left\{0+\frac{a^2}{2}{\sin }^{-1}(1)-0-0\right\}\\ \text{ =4b/a}(\frac{a^2}{2})(\pi /2)=\pi ab\text{ sq}\text{.units}\text{.}\end{array}$

Hence, Area of the Ellipse is $A$$=\pi ab\text{ sq}\text{. units}\text{.}$