Question

If the bounds of integration are not given when asked to solve a line integral, how do I solve for them?

Answer 1

\[y=\int\limits_a^b x \text d x= \left.\frac{x^2}{2}\right|_a^b\]
\[y=\int x \text d x= \frac{x^2}{2}+C\]

Answer 2

I think it depends on the curve you're integrating over. Otherwise, the bounds should be given.

Answer 3

you just add an arbitrary constant of integration which an element of the real numbers,
ive used C as this number,

Answer 4

For instance, this is what I have been given:

Evaluate the line integral
∫ xydx + y^2 dy + yzdz where C is the line segment from (1,0,−1) to (3, 4, 2).

So I parameterized the line segment where r_0=(1,0,-1) and r_1=(3,4,2) such that
R(t)=(1-t)(1,0,-1)+t(3,4,2) and solved for x, y,z and then dx,dy,dz to get it in terms of dt. I understand how to set up the integrand for a line integral, but the bounds are usually specified but for this problem they were not and I'm unsure if I am just supposed to integrate from 0<t<1 or if there is a way to solve for the bounds.

Answer 5

Ohh. I get it. In that case, t always runs from 0 to 1. :D

Answer 6

Why is that the case?

Let's say the line integral to be evaluated is in terms of spherical coordinates? Would I always set my bounds from 0 to Pi?

Answer 7

Any value of t<0 or t>1 will be outside the segment you parametrized.
And it only applies for line segments. :D