F dot dr problems

joemama69

Homework Statement

P = pi

Evaluate $$\int$$ F $$\cdot$$dr where c is the curve given by r(t) = (t+sin$$\pi$$t)i + (2tcos$$\pi$$t)j

F = (4x3y2 - 2xy3) i + (2x4y - 3x2y2 + 4y3)j

The Attempt at a Solution

When I dot them I get an extremely long expression.

$$\int$$ 4x3y2t - 4xy3t - 2xy3sinPt + 4x4yt + 2x4cosPt - 6x2y2t - 3x2y2cosPt + 8y3t + 4y3cosPt dt evaluated from t = 0 to to = 1

2x3y2 - 2xy3 +2Pxy3cosPt + 2x4y + 2Px4ysinPt - 3x2y2 - 3Px2y2sinPt + 4y3 + 4Py3sinPt

The Attempt at a Solution

Homework Helper
Gold Member
I haven'T checked it you have doted correctly but once that is done, you have to replace all the x in there by t+sin(pi*t) and all the y by 2tcos(pi*t). Then simplify if possible and integrate...

Homework Helper
F = (4x3y2 - 2xy3) i + (2x4y - 3x2y2 + 4y3)j

Don't even think of doing it directly!

What is the curl of F? Once you spot that, use Green's theorem or some other property to get the result in one line.

joemama69
Ah hah, Is this right

$$\int$$F dot dr = $$\int$$curl F dot dA = 0

Because

F = (4x3y2 - 2xy3) i + (2x4y - 3x2y2 + 4y3)j

curl F = (8x3y - 6xy2 - 8x3y + 6xy2)k = 0