Compute the line integral of F = <X^3, 4X> along the path from A to B. The path from A to B is not closed, it starts at A which has coordinates (-1,0) goes to (1,0) then goes up to (1,1) then left to (-2,1) then down to (-2,-1) and finally stops at B at (-1,-1).
Hint: To save work, use Green's Theorem to relate this time integral to the line integral along the vertical path from B to A.
To compute the line integral of the vector field F = <X^3, 4X> along the given path from A to B, we can use Green's Theorem to relate it to the line integral along a different path.
Green's Theorem states that for a vector field F = <P, Q> and a simple closed curve C, the line integral of F along C is equal to the double integral of the curl of F over the region R enclosed by C.
In this case, we are given that the path from A to B is not closed. However, we can still use Green's Theorem by considering a closed path formed by going from B to A along a vertical line. The vertical path will start at B and end at A.
First, let's find the line integral along the vertical path from B to A. The vertical line segment starts at point B (-1, -1) and ends at point A (-1, 0). We can parameterize this line segment as follows: x = -1 and y varies from -1 to 0.
Now, we need to compute the line integral of F along this vertical line segment. Recall that the line integral of a vector field F along a curve C can be computed as follows:
∫ F · dr = ∫ P dx + Q dy
Since P = X^3 and Q = 4X, we can substitute these values into the line integral:
∫ F · dr = ∫ (X^3) dx + 4X dy
Now, we need to parameterize the vertical line segment. Since x = -1 and y varies from -1 to 0, we can express x as a function of y:
x = -1
y varies from -1 to 0
Now, we can substitute these values into the line integral equation:
∫ F · dr = ∫ (-1^3) dx + 4(-1) dy
Simplifying this expression:
∫ F · dr = ∫ -1 dx - 4 dy
Now, we need to integrate with respect to the parameter y:
∫ F · dr = -x - 4y
Next, we need to evaluate this expression at the endpoints of the vertical line segment. At the starting point B (-1, -1), x = -1 and y = -1. At the ending point A (-1, 0), x = -1 and y = 0. Substituting these values into the expression:
∫ F · dr = -(-1) - 4(0) - (-1) - 4(-1) = 1 - 4 = -3
Therefore, the line integral along the vertical path from B to A is -3.
Now, to compute the original line integral along the path from A to B, we can use Green's Theorem. Green's Theorem states that the line integral of F along any curve C is equal to the double integral of the curl of F over the region R enclosed by C.
In this case, we are considering a clockwise-oriented closed path formed by the original path from A to B and the vertical path from B to A. We need to compute the double integral of the curl of F over the region enclosed by these two paths.
The curl of F can be computed as follows:
curl(F) = (∂Q/∂x) - (∂P/∂y)
Given that P = X^3 and Q = 4X, we can substitute these values into the curl equation:
curl(F) = (∂(4X)/∂x) - (∂(X^3)/∂y)
= 4 - 3X^2
Now, we need to compute the double integral of the curl of F over the region R enclosed by the two paths.
However, since the curve C is formed by two line segments, we can split the region R into two separate regions R1 and R2.
R1 is the region enclosed by the path from A to B, and R2 is the region enclosed by the vertical path from B to A.
Now, we can compute the double integral of the curl of F over each region. The integral of 4 over any region is the area of that region, which is simply given by the width times the height.
In region R1, the width is 2 (from x = -1 to x = 1) and the height is 0 (from y = 0 to y = 0), so the double integral of the curl of F over R1 is 4 * 2 * 0 = 0.
In region R2, the width is 0 (from x = -1 to x = -1) and the height is 1 (from y = -1 to y = 0), so the double integral of the curl of F over R2 is 4 * 0 * 1 = 0.
Therefore, the double integral of the curl of F over the region R enclosed by the two paths is 0 + 0 = 0.
According to Green's Theorem, the line integral of F along the path from A to B is equal to the double integral of the curl of F over the region R.
So, the line integral of F along the path from A to B is 0.
Therefore, the line integral of F = <X^3, 4X> along the path from A to B is 0.