Evaluate the line integrals $\int_C G(x,y) dx$, $\int_C G(x,y) dy$, and $\int_C G(x,y) ds$ on the indicated curve C. $$G(x,y) = 3x^2 +6y^2; y=2x+1, -1 \le x \le 0$$
I feel like this problem should be really easy, but I'm having a lot of trouble figuring it out. The answers at the back of my textbook are 3, 6, and $3\sqrt{5}$. If anyone could show me how to do this, it would be much appreciated.
Here's how to do the first one; hopefully this will give you some insight into how to attack the other two :)
Since the integral is with respect to $x$, it would be nice if we could parametrize $C$ with $x$. In fact this is the form in which it's given: $C = \{(x, 2x + 1) : -1 \leq x \leq 0\}$. Therefore:
$$\int_C G(x, y)dx \space = \int_{x = -1}^{x = 0} (3x^2 + 6(2x + 1)^2)dx \space = \space 3\int_{-1}^0 (9x^2 + 8x + 2)dx$$
You can finish it from there, and it does indeed equal 3. Hope this helps!