Answer:
To write an equivalent iterated integral using Green's theorem, we first need to find the curl of the vector field F = (ye^x, 2e^2).
The curl of a vector field F = (P, Q) is given by the formula ∇ x F = ∂Q/∂x - ∂P/∂y, where ∇ is the del operator. In this case, P = ye^x and Q = 2e^2.
Let's find the partial derivatives of Q and P with respect to x and y, respectively:
∂Q/∂x = 0 (since Q doesn't have an x component)
∂P/∂y = e^x (the derivative of ye^x with respect to y is 0, as y is not a function of y)
Now, using the formula for the curl, we have:
∇ x F = (0 - e^x)i + 0j = -e^x i
Next, we need to find the area enclosed by the given rectangle. The rectangle has a width of 5 and a height of 6, so the area is A = 5 * 6 = 30.
Finally, we can use Green's theorem to evaluate the line integral. Green's theorem states that for a vector field F = (P, Q) and a region R bounded by a positively oriented curve C, the line integral of F along C is equal to the double integral of the curl of F over the region R:
∫C ye^x dx + 2e^2 dy = ∬R -e^x dA
Since the region R is a rectangle, we can write the double integral as an iterated integral:
∬R -e^x dA = ∫0^6 ∫0^5 -e^x dx dy
Therefore, the equivalent iterated integral using Green's theorem is:
∫C ye^x dx + 2e^2 dy = ∫0^6 ∫0^5 -e^x dx dy