Use Green's theorem to evaluate the integral:

y^(2)dx+xy dy
where C is the boundary of the region lying between the graphs of y=0,
y=sqrt(x), and x=9

Using the definition, we have

P = y^2
Q = xy

and the integral becomes

∫[0,9]∫[0,√x] (∂Q/∂x - ∂P/∂y) dy dx
= ∫[0,9]∫[0,√x] (y - 2y) dy dx
= ∫[0,9]∫[0,√x] -y dy dx
= ∫[0,9] -x/2 dx
= -x^2/4 [0,9]
= -81/4

To use Green's theorem to evaluate the given line integral, we first need to determine the region enclosed by the curve C. The curve C is the boundary of the region lying between the graphs of y = 0, y = √(x), and x = 9.

Step 1: Determine the region of integration
The curve y = 0 represents the x-axis. The curve y = √(x) is a semicircular curve. And the line x = 9 is a vertical line. The region enclosed by these curves is a quarter circle with a radius of 9 units, located in the first quadrant.

Step 2: Parameterize the curve C

The curve C consists of three parts: part 1 is the x-axis from x = 0 to x = 9, part 2 is the curve y = √(x) from x = 9 to x = 0, and part 3 is the vertical line x = 9 from y = 0 to y = √(9).

The parameterization for part 1 (the x-axis) can be written as:
r1(t) = (t, 0), where t varies from 0 to 9.

The parameterization for part 2 (the curve y = √(x)) can be written as:
r2(t) = (t, √(t)), where t varies from 9 to 0.

The parameterization for part 3 (the vertical line x = 9) can be written as:
r3(t) = (9, t), where t varies from 0 to √(9).

Step 3: Compute the line integral using Green's theorem

Green's theorem states that for a vector field F = (P, Q) and a region R with a positively oriented, piecewise smooth boundary curve C parameterized by r(t) = (x(t), y(t)), the line integral of F along C can be evaluated as the double integral of the curl of F over the region R.

In this case,

F = (P, Q) = (y^2, xy)
Curl(F) = ∂Q/∂x - ∂P/∂y
= x - 2y

Using parameterizations for C, we can write dx = dx/dt dt and dy = dy/dt dt.

Therefore, the line integral can be written as:

∫(C) F · dr = ∫(R) (x - 2y) dA

where dA = dx dy = (dx/dt)(dy/dt) dt.

Since the region R is a quarter circle, we can set up our double integral as follows:

∫(C) F · dr = ∬(R) (x - 2y) dA
= ∫[0,9] ∫[0,√(x)] (x - 2y) dy dx

Step 4: Evaluate the double integral

∫[0,9] ∫[0,√(x)] (x - 2y) dy dx
= ∫[0,9] [(xy - y^2)]|[0,√(x)] dx
= ∫[0,9] (x√(x) - x) dx
= ∫[0,9] (x^(3/2) - x) dx

Evaluating this integral will give you the final answer for the given line integral using Green's theorem and the region enclosed by the curve C.