CBSE Class 12 · 2024 · Area Under CurvesLong Answer

Find the area of the region bounded by the curve

4 x^{2} + y^{2} = 36

using integration.

Answer: $18 π$ square units

Solution

Write the curve as

\frac{x ^{2}}{9} + \frac{y ^{2}}{36} = 1

, an ellipse with semi-axes

a = 3

(along

x

) and

b = 6

(along

y

).
The whole area is four times the first-quadrant area, where

y = 6 1 - \frac{x ^{2}}{9} = 2 9 - x^{2}

A = 4 \int_{0}^{3} 2 9 - x^{2} d x = 8 [\frac{x}{2} 9 - x^{2} + \frac{9}{2} sin^{- 1} \frac{x}{3}]_{0}^{3} = 8 \cdot \frac{9}{2} \cdot \frac{π}{2} = 18 π .

Hence the area enclosed by the ellipse is

18 π

square units.

CBSE Class 12 Board · 2024 · Subjective