Final answer:
Using Green's theorem, the value of the line integral ∮C 5xy dx + y^2 - 9 dy along the curve consisting of line segments from (0, 0) to (1, 1) to (0, 1) and back to (0, 0) is 0.
Step-by-step explanation:
To evaluate the line integral using Green's theorem, we need to compute the line integral of the given function along the curve. The curve consists of line segments from (0, 0) to (1, 1), then from (1, 1) to (0, 1), and finally from (0, 1) back to (0, 0).
We can divide the curve into two segments. The first segment from (0, 0) to (1, 1) can be parameterized as x = t, y = t for t in [0, 1]. The second segment from (1, 1) to (0, 1) can be parameterized as x = 1 - t, y = 1 for t in [0, 1].
Using Green's theorem, we have:
∮C 5xy dx + y^2 - 9 dy = ∫∫R (partial derivative of y^2 - 9 with respect to x - partial derivative of 5xy with respect to y) dA
where R is the region enclosed by the curve C.
For the first segment, we have:
∫∫R1 (∂/∂x(y^2-9) - ∂/∂y(5xy)) dA = ∫∫R1 (-5y) dA
where R1 is the region enclosed by the first segment of the curve.
For the second segment, we have:
∫∫R2 (∂/∂x(y^2-9) - ∂/∂y(5xy)) dA = ∫∫R2 (10x) dA
where R2 is the region enclosed by the second segment of the curve.
Calculating these double integrals, we get:
∫∫R1 (-5y) dA = -5∫∫R1 y dA = -5 * area of R1
∫∫R2 (10x) dA = 10∫∫R2 x dA = 10 * area of R2
Since R1 is a triangle with base 1 and height 1, the area of R1 is 1/2. Similarly, R2 is also a triangle with base 1 and height 1, so the area of R2 is 1/2.
Therefore, the value of the line integral ∮C 5xy dx + y^2 - 9 dy is:
-5 * (1/2) + 10 * (1/2) = 0.