Final answer:
The line integral xy dx + x^2 dy around the given rectangle is evaluated by breaking the path into four segments. After computing the integral for each segment, the total value comes out to be 20.
Step-by-step explanation:
The question involves evaluating a line integral directly for a rectangular path in a vector field. Specifically, we are given the integral xy dx + x2 dy, where C is the path counterclockwise around the rectangle with vertices (0, 0), (2, 0), (2, 5), (0, 5). To evaluate this, we'll break down the path into four segments and integrate over each segment separately.
For the bottom segment (0,0) to (2,0), y=0 and dy=0, so the integral is zero. For the right segment (2,0) to (2,5), x=2, thus the integral becomes ∫ x2 dy over y from 0 to 5, which is 22 ∫ dy equal to 4y evaluated from 0 to 5, giving us 20. For the top segment (2,5) to (0,5), y=5 and dy=0, and the integral is again zero, since dx changes direction and the values negate each other. Lastly, for the left segment (0,5) to (0,0), x=0, thus the integral is zero because x2 is zero.
Add up the integrals over each segment, we get 0 + 20 + 0 + 0, resulting in a total line integral value of 20.