CBSE Class 12 · 2019 · Area Under CurvesLong Answer

Using method of integration, find the area of the region enclosed between two circles

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

and

(x - 2)^{2} + y^{2} = 4

Answer: $2 (\frac{4 π}{3} - 3) = (\frac{8 π}{3} - 23)$ square units

Solution

Both circles have radius

2

, centres

O (0, 0)

and

C (2, 0)

; eliminating

y

gives

x = 1

, so they meet at

(1, \pm 3)

.
By symmetry about the

x

-axis,

A = 2 [\int_{0}^{1} 4 - (x - 2)^{2} d x + \int_{1}^{2} 4 - x^{2} d x] .

Evaluating the standard integrals,

A = 2 (\frac{4 π}{3} - 3) = \frac{8 π}{3} - 23 square units.

CBSE Class 12 Board · 2019 · Subjective