For the vector field G⃗ =(yexy+3cos(3x+y))i⃗ +(xexy+cos(3x+y))j⃗ , find the line integral of G⃗ along the curve C from the origin along the x-axis to the point (3,0) and then counterclockwise around the circumference of the circle x2+y2=9 to the point (3/2‾√,3/2‾√).

How would I parameterize the line integral?

along the line from (0,0) to (3,0), dy=0, so that part is just

∫[0,3] G(x,0) dx

For the 2nd part, use the usual polar coordinate conversion with r=3, and you have

∫[0,π/4] G(3cosθ,3sinθ) dθ

I can't tell what your radicals are supposed to be. √3/2?

The line integral can be calculated using the Fundamental Theorem of Line Integrals b/c

i and j vectors are integrals of e^(xy) + sin(x+y)...

Integral of that from (0,0) to ( (3/sqrt(2)) , (3/sqrt(2)) )

To parameterize the line integral, we need to express the curve C as a set of vectors with a parameter.

For the first part of the curve, which goes from the origin along the x-axis to the point (3,0), we can use the parameter t and vary it from 0 to 3.

For this section, the curve is a straight line, so its equation can be written as r(t) = (x(t), y(t)), where x(t) = t and y(t) = 0.

So, the parameterization of the first part of the curve is:

r(t) = (t, 0), for t in [0, 3].

For the second part of the curve, which goes counterclockwise around the circumference of the circle x^2 + y^2 = 9 from the point (3,0) to the point (3/2√2, 3/2√2), we can use the parameter θ and vary it from 0 to π/4.

The equation of the circle can be expressed in polar coordinates as r(θ) = (x(θ), y(θ)), where x(θ) = 3cos(θ) and y(θ) = 3sin(θ).

So, the parameterization of the second part of the curve is:

r(θ) = (3cos(θ), 3sin(θ)), for θ in [0, π/4].

To summarize, the parameterizations for the line integral are:

For the first part: r(t) = (t, 0), for t in [0, 3].
For the second part: r(θ) = (3cos(θ), 3sin(θ)), for θ in [0, π/4].

To parameterize the line integral, we need to define a parameterization for each segment of the curve separately.

1. Along the x-axis from the origin to the point (3, 0):
Since the curve is a straight line along the x-axis, we can parameterize it as:
x(t) = t, y(t) = 0, where t ranges from 0 to 3.

2. Along the circumference of the circle x^2 + y^2 = 9:
To parameterize the circle, we can use polar coordinates:
x(t) = 3cos(t), y(t) = 3sin(t), where t ranges from 0 to π/2.

Therefore, the parameterization for the line integral along the entire curve C consists of two segments:
1. Along the x-axis: x(t) = t, y(t) = 0, where t ranges from 0 to 3.
2. Along the circle: x(t) = 3cos(t), y(t) = 3sin(t), where t ranges from 0 to π/2.

By combining the parameterizations of both segments, you can evaluate the line integral of the vector field G⃗ along the curve C.