Mathematical Methods for Physics and Engineering : A Comprehensive Guide

2.2 INTEGRATION

dA

ρ(φ)

ρ(φ+dφ)

ρdφ

x

y

O

B

C

Figure 2.9 Finding the area of a sectorOBCdefined by the curveρ(φ)and

the radiiOB,OC, at angles to thex-axisφ 1 ,φ 2 respectively.

dA=^12 ρ^2 dφ, as illustrated in figure 2.9, and hence the total area between two

anglesφ 1 andφ 2 is given by

A=

∫φ 2

φ 1

1

2 ρ

(^2) dφ. (2.38)
An immediate observation is that the area of a circle of radiusais given by
A=
∫ 2 π
0
1
2 a
(^2) dφ=[ 1
2 a
(^2) φ]^2 π
0 =πa
(^2).

The equation in polar coordinates of an ellipse with semi-axesaandbis
1
ρ^2

=

cos^2 φ

a^2

+

sin^2 φ

b^2

.

Find the areaAof the ellipse.

Using (2.38) and symmetry, we have

A=

1

2

∫ 2 π

0

a^2 b^2

b^2 cos^2 φ+a^2 sin^2 φ

dφ=2a^2 b^2

∫π/ 2

0

1

b^2 cos^2 φ+a^2 sin^2 φ

dφ.

To evaluate this integral we writet=tanφand use (2.35):

A=2a^2 b^2

∫∞

0

1

b^2 +a^2 t^2

dt=2b^2

∫∞

0

1

(b/a)^2 +t^2

dt.

Finally, from the list of standard integrals (see subsection 2.2.3),

A=2b^2

[

1

(b/a)

tan−^1

t

(b/a)

]∞

0

=2ab

(π

2

− 0

)

=πab.