Green's Theorem & Line Integral confusion

[Dead85]

Homework Statement

a) Evaluate the work done by the force field F(x, y) = (3y^(2) + x)i + 4x^(3)j over the curve
r(t) = e^(t)i + e^(3t)j, tε[0, ln(2)].
b) Using Green’s theorem, find the area enclosed by the curve r(t) and the segment that
joins the points (1, 1) and (2, 8).
c) Find the flux of F across the curve described in b).

Homework Equations

I may be missing something but for the life of me I can't figure out how to answer part b.). I already have part a.) and can do part c.) just need to figure out the limits for part be.

The Attempt at a Solution

a.)∫(0 to ln2 )[3e^(7t) + e^(2t) + 12e^(7t)]dt= 2547

B.)Greens Theorem
∫(12x^2-(6y+1))dA

Any help would be awesome!

 

[DryRun]

For part (b): Draw the graph of the curve r(t) in the given interval. You only need to plot 3 points to get a general idea of the shape of the graph. Try the following values of t: 0, ln (1) and ln (2). Then, plot the line that joins the points (1,1) and (2,8). Find its equation. Then describe the enclosed region and find its area using the Green's theorem.

 

[tiny-tim]

Welcome to PF!

Hi Dead85! Welcome to PF!

(try using the X² button just above the Reply box )

r(t) = e^ti + e^3tj, tε[0, ln(2)].
b) Using Green’s theorem, find the area enclosed by the curve r(t) and the segment that
joins the points (1, 1) and (2, 8).

So r is part of y = x³.

To find the area, you need ∫∫ 1 dA.

So to use Green's theorem, you need a function (G(x,y),H(x,y)) with ∂H/∂x - ∂G/∂y = 1.

Try something like (0,-x) or (y,0). ​