I am given a problem as follows,
Compute the length of the helix that wraps 5 times around the lateral side of a right circular cylinder of radius R and height H w/ a constant pitch (so each wrap rises the same direction up the cylinder)
The answer: Sqrt[ (10π R^2) + H^2 ]
My issue: Conceptually, so far we have been computing integrals over curves through breaking the curves into small pieces of the length ds (w/ is the magnitude of dr= (dx)i + (dy)j). Where other problems were straight forward given an equation, all I had to do was describe the length element in terms of one variable, however, I'm not sure how that problem is depicted in this problem, i do understand how if we were to set the integral up, it's bounds would be 0
Basically I'm looking for the framework behind coming up with this answer as can best be done on Yahoo answers....lol thanks in advance math friends
Compute the length of the helix that wraps 5 times around the lateral side of a right circular cylinder of radius R and height H w/ a constant pitch (so each wrap rises the same direction up the cylinder)
The answer: Sqrt[ (10π R^2) + H^2 ]
My issue: Conceptually, so far we have been computing integrals over curves through breaking the curves into small pieces of the length ds (w/ is the magnitude of dr= (dx)i + (dy)j). Where other problems were straight forward given an equation, all I had to do was describe the length element in terms of one variable, however, I'm not sure how that problem is depicted in this problem, i do understand how if we were to set the integral up, it's bounds would be 0
Basically I'm looking for the framework behind coming up with this answer as can best be done on Yahoo answers....lol thanks in advance math friends
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This is more like a Calc 3 problem.
I will use variable t instead of x
the helix equation is:
r(t) =
L = ∫ || r(t) || dt, t = 0 to 5(2pi)
L = ∫ sqrt[ (-Rsint)^2 + (R cost)^2 + ( H/(10pi))^2 ] dt
L = ∫ sqrt[ R^2 + (H/(10pi))^2 ] dt
L = sqrt[ R^2 + (H/(10pi))^2 ] t
evaluating the limits
L = sqrt[ R^2 + (H/(10pi))^2 ] (10pi)
L = sqrt ( (10pi R)^2 + H^2 )
Perhaps, you had a typo. You had 10pi (R^2) while I had (10pi R)^2
I will use variable t instead of x
the helix equation is:
r(t) =
L = ∫ || r(t) || dt, t = 0 to 5(2pi)
L = ∫ sqrt[ (-Rsint)^2 + (R cost)^2 + ( H/(10pi))^2 ] dt
L = ∫ sqrt[ R^2 + (H/(10pi))^2 ] dt
L = sqrt[ R^2 + (H/(10pi))^2 ] t
evaluating the limits
L = sqrt[ R^2 + (H/(10pi))^2 ] (10pi)
L = sqrt ( (10pi R)^2 + H^2 )
Perhaps, you had a typo. You had 10pi (R^2) while I had (10pi R)^2