00:01
For this problem, we are asked to find the value of the integral across curve c of x squared plus y squared dx plus 2x -y -d -y -d -y.
00:09
So the first thing that i'm going to do here is check to see if our vector field we're integrating here is conservative.
00:16
We take the partial derivative with respect to y of x squared plus y squared, which gives us 2y.
00:23
Then we take the partial derivative with respect to x of 2xy, which gives us 2y, confirming that, that this is a conservative vector field, therefore path independent.
00:35
So for part a, we can re -parameterize into something easier.
00:39
You can see that we'd go from the point zero -zero to the point -eight -four, so we can parameterize this simply as 8t, 4t, for t between 0 and 1.
00:58
Then we'll have that dx is going to equal 8d, and d .y will equal 4d, d .t, which then means that our integral of f .dr here will become, so when we substitute everything in, we can rewrite our line integral as the integral from zero to one of simply 576 t squared d t, which will then become 576 times t cubed over three, oops, times t cubed over three evaluated from zero to one, which will give us a final result of 5706 times 575...