Use Green's Theorem to evaluate
F(x, y) = y cos x − xy sin x, xy + x cos x
, C is the triangle from (0, 0) to (0, 4) to (2, 0) to (0, 0)
I kind of understand how to do this , but I am having trouble with the trig
what does the comma mean in
F(x, y) = y cos x − xy sin x, xy + x cos x
?
are we to integrate F around the perimeter of the triangle by integrating the divergence over the area or what ?
Look at earlier questions below.
To evaluate the line integral using Green's Theorem, we need to perform a double integration over the region enclosed by the given curve. Let's break down the steps to apply Green's Theorem:
1. Determine the curl of the vector field F(x, y):
The curl of a vector field F(x, y) = P(x, y)i + Q(x, y)j is given by:
curl(F) = ∂Q/∂x − ∂P/∂y
Here, P(x, y) = y cos(x) − xy sin(x) and Q(x, y) = xy + x cos(x).
To find ∂Q/∂x, differentiate Q(x, y) with respect to x and simplify:
∂Q/∂x = y + cos(x) − y sin(x)
To find ∂P/∂y, differentiate P(x, y) with respect to y and simplify:
∂P/∂y = cos(x) − x sin(x)
Now, we have the curl of the vector field F(x, y):
curl(F) = (y + cos(x) − y sin(x))i + (cos(x) − x sin(x))j
2. Determine the area enclosed by the curve:
The given curve is a triangle with vertices (0, 0), (0, 4), and (2, 0). To calculate the area of this triangle, we can use the formula for the area of a triangle:
Area = 1/2 * base * height
Here, the base of the triangle is the distance between (0, 0) and (2, 0), which is 2 units. The height is the distance between (0, 0) and (0, 4), which is 4 units.
Area = 1/2 * 2 * 4 = 4 square units
3. Apply Green's Theorem:
According to Green's Theorem, the line integral of a vector field F around a closed curve C is equal to the double integral of the curl of F over the region R enclosed by C.
Line integral of F around C = ∬R curl(F) · dA
Since the area enclosed by the curve is a triangle, we can rewrite the double integral as an iterated integral:
Line integral of F around C = ∫[y = 0 to 4] ∫[x = 0 to 2] (y + cos(x) − y sin(x)) dx dy + ∫[y = 0 to 4] ∫[x = 0 to 2] (cos(x) − x sin(x)) dy dx
4. Evaluate the double integrals:
To evaluate the double integrals, we integrate the individual terms with respect to x and y, following the limits provided in the iterated integrals:
Line integral of F around C = ∫[y = 0 to 4] [(xy + sin(x) + y^2/2) from x = 0 to 2] dy
- ∫[y = 0 to 4] [sin(2) - sin(y)] dy
Simplifying and evaluating the integrals, we get:
Line integral of F around C = [(2y + 2sin(2) + y^2) from y = 0 to 4] - [4sin(2) + cos(4) - cos(0)]
Line integral of F around C = (8 + 2sin(2) + 16 + 16) - (4sin(2) + cos(4) - 1)
Line integral of F around C = 40 + 2sin(2) - 4sin(2) + cos(4) - 1
Line integral of F around C = 39 + 2sin(2) + cos(4)
So, the value of the line integral using Green's Theorem is 39 + 2sin(2) + cos(4).