Let F(x,y) = xy^2 i + x^2y j (a) Find a function f such that F = (gradient)f (b)

Question

Let F(x,y) = xy^2 i + x^2y j

(a) Find a function f such that F = (gradient)f

(b) Use part (a) to evaluate the integral of [(gradient)f * dr] along any smooth curve from (1,1) to (2,2).

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For part a, I integrated the 'i' portion of the function and found F = (x^2y^2)/2
I then differentiated with respect to y and found fy = x^2y
Since I can get i and j by differentiating F with respect to fx and fy, is this what the question is asking for?

For part b, I did a double integral: Int[1_2] [Int[1_2](x^2y^2/2) ] and ended up with 48/18?

Please show your work so I can follow along. Thanks!

Explanation / Answer

a) f= (x^2y^2)/2i+(x^2y^2)/2j

b) integral of [gradf*dr]=integral of [F*dr]

parametrize the integral as x=(1-t)x1+tx2 and y=(1-t)y1+ty2

this gives x=y=1+t dx=dy=dt

F.dr=(xy^2 i + x^2y j).(dxi+dyj)

=2*(1+t)^3

integrate it from 0 to 1 with respect to t, get 15/2

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