is a vector field. Now how do I evaluate the line integral F.dr in the region C where C starts at the point A(12,0,0), ends at point B (0,12,5) and is an arc on the helix:
where π is the number pi.
I know how to evaluate line integrals for a helix and using F but now with the points added in (point A and B) I have no idea how to begin. Please help.
First, find the limits of integration. That is, find the start and endpoints of your path.
A = (12,0,0) B = (0,12,5)
So we want (12cos(pi*t), 12sin(pi*t), 2t) = (12, 0, 0). So it must be that 2t = 0. Thus, t = 0. Try the other ones, and you'll see that they work as well.
Now we want (12cos(pi*t), 12sin(pi*t), 2t) = (0, 12, 5). Again, 2t = 5, so t = 5/2. You can check the others, but it will work out.
So we know that 0 <= t <= 5/2. These are the limits of integration.
So we need to find F(c(t)) and c'(t). Once you find these, take the dot product, and then integrate.
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