Multivariate Calculus - Bruce E. Shapiro

40 LECTURE 5. VELOCITY, ACCELERATION, AND CURVATURE
Example 5.3 Find the length of the curve r(t) = (t cos t, t sin t, t) over the interval
(3, 4).
Solution. Differentiating,
hence
s =
=
=
=
=
∫ 4
3
∫ 4
3
∫ 4
3
∫ 4
3
∫ 4
3
r ′ (t) = (−t sin t + cos t, t cos t + sin t, 1)
√
(−t sin t + cos t) 2 + (t cos t + sin t) 2 + (1) 2 dt
√
t 2 sin 2 t − 2t sin t cos t + cos 2 t + t 2 cos 2 t + 2 sin t cos t + sin 2 t + 1 dt
√
t 2 sin 2 t + cos 2 t + t 2 cos 2 t + sin 2 t + 1 dt
√
t 2 (sin 2 t + cos 2 t) + (cos 2 t + sin 2 t) + 1 dt
√ ∫ 4 √
t 2 + 1 + 1 dt = t 2 + 2dt
3
By formula 44 on the inside back cover of the text
∫ √x 2 ± a 2 dx = x 2
√
x ± a 2 ± a2
2 ln ∣ ∣∣x +
√
x 2 ± a 2 ∣ ∣∣
so that
s =
=
=
≈
∫ 4 √
( t √ t 2 + 2 dt = t
3
2
2 ∣
+ 2 + ln ∣t + √ )∣
t 2 ∣∣∣
4
+ 2∣
3
(
2 √ 4 2 ∣
+ 2 + ln ∣4 + √ ) (
4 2 3 √
+ 2∣
− 3
2
2 + 2 + ln |3 + √ )
3 2 + 2|
(
2 √ ∣
18 + ln ∣4 + √ ) ( 3
∣ ∣∣3 √ ∣)
∣∣
18∣
− 11 + ln + 11
2√
8.485 + 2.109 − 4.975 − 1.843 ≈ 3.776.
Revised December 6, 2006. Math 250, Fall 2006