Calculate the following line integral

[lep11]

Homework Statement

Let $f(x,y)=(xy,y)$ and $\gamma:[0,2\pi]\rightarrowℝ^2$,$\gamma(t)=(r\cos(t),r\sin(t))$, $r>0$. Calculate $\int_\gamma{f{\cdot}d\gamma}$.

Homework Equations

The Attempt at a Solution

The answer is 0. Here's my work. However, this method requires that you are familiar with some useful trig identities.

Could someone please take a look at it and check if it's correct? Are there alternative ways? I have also tried to find the potential function $u$, $\nabla{u}=f$...

 

[Ray Vickson]

Homework Statement

Let $f(x,y)=(xy,y)$ and $\gamma:[0,2\pi]\rightarrowℝ^2$,$\gamma(t)=(r\cos(t),r\sin(t))$, $r>0$. Calculate $\int_\gamma{f{\cdot}d\gamma}$.

Homework Equations

The Attempt at a Solution

The answer is 0. Here's my work. However, this method requires that you are familiar with some useful trig identities.

Could someone please take a look at it and check if it's correct? Are there alternative ways? I have also tried to find the potential function $u$, $\nabla{u}=f$...

I get an answer of 0 as well, but I have not checked your work because I do not look at posted images, but only at typed versions.