I am having trouble with a homework problem from Calc 3. I am not sure how to approach it. In is in the section for line integrals, but the fundamental thm for line integrals has not been introduced yet. This whole chapter has been diffult at times to understand what is being done with the integrals.
If C is a smooth curve given by a vector function r(t), a≤t≤b, and v is a constant vector, show
Integral v.dr = v.[r(b)-r(a)]
The only thing I can think of is to take the integral of v(r(t)).r'(t) dt evaluated from a to b then that equals Integral of V. T ds but I have no idea how to get it to equal v.(r(b)-r(a))
Re: line integral
The key here is that if the vector is constant, there is no need for an integral. Its kind of like when you are integrating a constant function in elementary calculus, you just multiply the value of the function times the change in x.
The same thing applies here. r(b)-r(a) is simply the displacement vector from your start to your end point, and you dot product it with the vector v. Basically since the vector field (v) is constant, you can just have 1 piece in the sum. The dr will be the displacement vector.
You might need to show that the path is irrelevant for constant fields, but that depends on what you are allowed to use.