Question

1)Sketch the vector field F⃗ (x,y)=xi⃗ +yj⃗ and calculate the line integral of F⃗ along the line segment from (2,5) to (2,9).help

Answer 1

It shows in the type section f and then a square, is that meant to be a symbol of some sort?

Answer 2

So you need some sort of parametrization of your line segment. Anything will do, preferably simple. The x value is constant and the y value increases from 5 to 9. So an example of a parametrization would be \(r(t) = 2i + (5+4t)j\) \(\ 0 \le x \le 1\). So the line integral is:
\[\int\limits_{C}^{}f(g(t),h(t))|r'(t)|dt\] So r'(t) = 4j, meaning |r'(t)| is simply 4. Plugging everything in we have:
\[\int\limits_{0}^{1}(2 + (5+4t))*4dt \implies \int\limits_{0}^{1}(28+16t)dt\] I'm sure you could take it from there :)
As for sketching the vector field. The vector field kind of pins vectors onto coordinate points. So if you plug in the point (1,1), the vector i + j gets anchored to the point (1,1). And the pattern continues, plugging in (-1,2) anchors the vector -i + 2j to the point (-1,2). Just keeping plotting vectors in that manner until you hae a sufficient amount.