I have an example problem of a line integral below:
It makes sense but the above problem is somewhat easy because the parametric equation is given to us.
I have this next problem:
- Calculate $\int_{C} F(r) \cdot dr$ for the given data.
$$F = [y^{2}, -x^{2}]$$
and C is a line from (0, 0) straight to (1, 4).
What is r(t)? CAn I say it's:
$$r(t) = [t, 4t]$$ $$r'(t) = [1, 4]$$ $$F(r(t)) = [16t^{2}, -t^{2}]$$
So the line integral is: $\int_0^1 [16t^{2}, -t^{2}] \cdot [1, 4]$
Is that right so far?
$$\int_0^1 [16t^{2} - 4t^{2}] $$ $$\int_0^1 [12t^{2}] $$
$$\left[4t^{3}\right]_0^1 = 4$$
Is that right?
Is the best way to interpret this the F's work done on something that is displaced along the straight line C by F's vectors along C?