Name: Area Enclosed by a Curve and a Line: Illustration
Uploaded: 2019-07-19
Duration: 3 min 35 s
Description: How would you find the area enclosed between a parabola and a straight line? Watch the video to see the method as an application of definite integrals.

Question

Find the area of the smaller region bounded by the ellipse

\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1

and the line

\frac{x}{a} + \frac{y}{b} = 1

.

Accepted Answer

Given: The equation of the ellipse: $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ and the line $\frac{x}{a} + \frac{y}{b} = 1$
The area of the smaller region bounded by the ellipse, $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ and the line, $\frac{x}{a} + \frac{y}{b} = 1$ is represented by the shaded region $B C A B$ as

The points of intersection after solving the two given equations: $A (0, b)$ and $B (a, 0)$

$\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ $\Rightarrow y = b \sqrt{1 - \frac{x^{2}}{a^{2}}} = y_{1}$

$\frac{x}{a} + \frac{y}{b} = 1$ $\Rightarrow y = b (1 - \frac{x}{a}) = y_{2}$

$∴ Area o f B C A B = Area o f (O B C A O) - Area o f (O B A O)$

$= \int_{0}^{a} y_{1} d x - \int_{0}^{a} y_{2} d x$

$= \int_{0}^{a} b \sqrt{1 - \frac{x^{2}}{a^{2}}} d x - \int_{0}^{a} b (1 - \frac{x}{a}) d x$

$= \frac{b}{a} \int_{0}^{a} \sqrt{a^{2} - x^{2}} d x - \frac{b}{a} \int_{0}^{a} (a - x) d x$

$= \frac{b}{a} [{\{\frac{x}{2} \sqrt{a^{2} - x^{2}} + \frac{a^{2}}{2} \sin^{- 1} \frac{x}{a}\}}_{0}^{a} - {\{a x - \frac{x^{2}}{2}\}}_{0}^{a}]$

Area of $B C A B = \frac{b}{a} [{\{\frac{x}{2} \sqrt{a^{2} - x^{2}} + \frac{a^{2}}{2} \sin^{- 1} \frac{x}{a}\}}_{0}^{a} - {\{a x - \frac{x^{2}}{2}\}}_{0}^{a}]$

$= \frac{b}{a} [\{\frac{a^{2}}{2} (\frac{π}{2})\} - \{a^{2} - \frac{a^{2}}{2}\}]$

$= \frac{b}{a} [\frac{a^{2} π}{4} - \frac{a^{2}}{2}]$

$= \frac{b a^{2}}{2 a} [\frac{π}{2} - 1]$

$= \frac{a b}{2} [\frac{π}{2} - 1]$

$= \frac{a b}{4} (π - 2)$ square units

Hence, the required area is $\frac{a b}{4} (π - 2)$ square units.

Find the area of the smaller region bounded by the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ and the line $\frac{x}{a} + \frac{y}{b} = 1$ .

Important Questions on Application of Integrals

Learn and Prepare for any exam you want !

Find the area of the smaller region bounded by the ellipse x2a2+y2b2=1 and the line xa+yb=1.

Important Questions on Application of Integrals

Learn and Prepare for any exam you want !

Find the area of the smaller region bounded by the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ and the line $\frac{x}{a} + \frac{y}{b} = 1$ .