
6-36 Calculus – Differentiation and Integration
Q(r + ∆r, θ + ∆θ)
∆θ
r + ∆r
θ
P(r, θ)
r
φ
ψ
O
T
M
x
Figure 6.16
Dividing the above expression by (Dq)
2
, we get
chord sin cosPQ
r
r
r
∆
=
∆
∆
+
∆
∆
+
− ∆
∆
q
q
q q
q
q
2
2
2
1
2
.
Thus,
∆
∆
=
∆
∆
s s
PQ
PQ
q q
2
2
chord
chord
=
∆
∆
+
∆
∆
+
2
2
2
2
arc
chord
sinPQ
PQ
r
rq
q q
rr
1
2
− ∆
∆
cos q
q
lim lim
arc
chord
lim
∆ → ∆ →
∆
∆
=
q q
q
0
2
0
2
s PQ
PQ
∆∆ →
∆
∆
+
∆
∆
+
− ∆
∆
q
q
q q
q
q
0
2
2 2
1
r
r
r
sin cos
= +
=
∆ →
r
dr
d
PQ
PQ
2
2
0
1
q
q
, lim
arc
chord
.
∴
= +
= +
ds
d
r
dr
d
ds
d
r
dr
d
q q
q q
2
2
2
2
i.e.,
2
.
Example 6.31 Find ...