Find the area of the smaller region bounded by the ellipse and the line contexto.140 3 years ago Solution: Area of the smaller region bounded by the ellipse, , and the line, is represented by the shaded region BCAB as
![\begin{array}{l} \therefore \text { Area } B C A B=\text { Area }(O B C A O)-\text { Area }(O B A O) \\ =\int_{0}^{a} b \sqrt{1-\frac{x^{2}}{a^{2}}} d x-\int_{0}^{a} b\left(1-\frac{x}{a}\right) d x \\ =\frac{b}{a} \int_{0}^{n} \sqrt{a^{2}-x^{2}} d x-\frac{b}{a} \int_{0}^{a}(a-x) d x \\ =\frac{b}{a}\left[\left\{\frac{x}{2} \sqrt{a^{2}-x^{2}}+\frac{a^{2}}{2} \sin ^{-1} \frac{x}{a}\right\}_{0}^{a}-\left\{a x-\frac{x^{2}}{2}\right\}_{0}^{a}\right] \\ =\frac{b}{a}\left[\left\{\frac{a^{2}}{2}\left(\frac{\pi}{2}\right)\right\}-\left\{a^{2}-\frac{a^{2}}{2}\right\}\right] \\ =\frac{b}{a}\left[\frac{a^{2} \pi}{4}-\frac{a^{2}}{2}\right] \\ =\frac{b a^{2}}{2 a}\left[\frac{\pi}{2}-1\right] \\ =\frac{a b}{2}\left[\frac{\pi}{2}-1\right] \\ =\frac{a b}{4}(\pi-2) sq. units \end{array}](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-52e3a925f62d316ccf983ad72b16a3de_l3.png "Rendered by QuickLaTeX.com")
