# Thread: length of curve

1. ## length of curve

find the length of the curve r(t) = tcost i + tsint j + sqrt(2)t k ,
0 <= t <= 1

help with this one please.

2. Originally Posted by rcmango
find the length of the curve r(t) = tcost i + tsint j + sqrt(2)t k ,
0 <= t <= 1

help with this one please.
r(t) = <t*cos(t), t*sin(t), sqrt(2)*t>

r'(t) = <cos(t) - t*sin(t), t*cos(t) + sin(t), sqrt(2)>

arclength = int(sqrt[(cos(t) - t*sin(t))^2 + (t*cos(t) + sin(t))^2 + 2]dt)

I'll let you do the simplifying, but it turns out to simplify to:

int(sqrt(t^2 + 3)dt) = (3*ln(sqrt(t^2 + 3) + t))/2 + (t*sqrt(t^2 + 3))/2

From t = 0...1, we have:

3*ln(3)/4 + 1 ~ 1.824