Find the work done by the force field F= 3yi - x^2j in moving a particle along the curve y=x^(1/2) from (1,1) to (4,2) so i parameterize and get x=1+3t, y= 1+t so 3(1+t)-(1+t)^2 then take the derivative of x and y and multiple to ge 9(1+t)-(1+t)^2 and i integrate that from 0-1 and i get 13/2 but the answer should be 39/5 so my question is where am i going wrong
The work done is given by: [tex] W=\int_{\gamma}\mathbf{F}\cdot d\mathbf{r} [/tex] where [itex]d\mathbf{r}=dx\mathbf{i}+dy\mathbf{j}[/itex]
This is wrong. If y= 1+ t, then t= y- 1 so x= 1+ 3(y- 1)= 3y- 2. That is linear- it is the straight line through (1, 1) and (4, 2). Use [itex]x= t^2[/itex], [itex]y= t[/itex], with t from 1 to 2 instead. Now F= 3ti- t^4 j, dx= 2tdt, dy= dt so [tex]\int_{t= 1}^2 (3t)(2t)dt- (t^4)dt= \int_{t=1}^2 (6t^2- t^4)dt[/tex]
Thanks Halls, that made sense. I have one more question how do you know to to parameterize it the way you did? is it because y=x^1/2 so choose x=t^2 ad y=t
Yes. Of course you could always use [itex]x= t[/itex], [itex]y= t^{1/2}[/itex] with t from 1 to 4. There are many different ways to parameterize any curve.