Practicing Success
The area of the smaller region bounded by the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$ and the line $\frac{x}{3}+\frac{y}{2}=1$ is : |
$\frac{2}{3}(π-2)$ $\frac{2}{3}(π+2)$ $\frac{3}{2}(π-2)$ $\frac{3}{2}(π+2)$ |
$\frac{3}{2}(π-2)$ |
Area of the smaller region bounded by the ellipse $\frac{x^{2}}{9}+\frac{y^{2}}{4}=1$, and line $\frac{x}{3}+\frac{y}{2}=1$ Represented by shaded region BCAB. ∴ Area BCAB = Area (OBCAO) – Area (OBAO) $=\int_{0}^{3} 2 \sqrt{1-\frac{x^{2}}{9}} d x-\int_{0}^{3} 2\left(1-\frac{x}{3}\right) d x$ $=\frac{2}{3}\left[\int_{0}^{3} \sqrt{9-x^{2}} d x\right]-\frac{2}{3} \int_{0}^{3}(3-x) d x$ $=\frac{2}{3}\left[\frac{x}{2} \sqrt{9-x^{2}}+\frac{9}{2} \sin ^{-1} \frac{x}{3}\right]_{0}^{3}-\frac{2}{3}\left[3 x-\frac{x^{2}}{2}\right]_{0}^{3}$ $=\frac{2}{3}\left[\frac{9}{2}\left(\frac{\pi}{2}\right)\right]-\frac{2}{3}\left[9-\frac{9}{2}\right]$ $=\frac{2}{3}\left[\frac{9 \pi}{4}-\frac{9}{2}\right]$ $=\frac{2}{3} \times \frac{9}{4}(\pi-2)$ $=\frac{3}{2}(\pi-2)$ units |