# F dot dr problems

## 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

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quasar987
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...

Shooting Star
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.

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