Use Green's Theorem to find the work done by the force $\mathbf{F}(x,y)=x(x+y)\mathbf{i}+xy^2\mathbf{j}$ in moving a particle from the origin along the $x$-axis to $(1,0)$, then along the line segment to $(0,1)$, and back to the origin along the $y$-axis.
So I was able to find $\frac{\partial Q}{\partial x} -\frac{\partial P}{\partial y}$ to be $y^2 -x$ and I integrated that with respect to $y$ and $x$ by using $y= 1-x$ as my upper bound and $y=0$ as my lower bound, and $0 < x < 1$ for my $x$ integral. but it came out to $-\frac{7}{36}$, and the answer is $-\frac1{12}$. I'm not sure if I'm doing something fundamentally wrong here or if its a calculation error. I checked it twice. How do I do it correctly?
Best Answer
Yes, you are correct, by Green's Theorem, you should evaluate $$\int_{x=0}^1\int_{y=0}^{1-x}(y^2-x)dydx=\int_{x=0}^1\left[\frac{y^3}{3}-xy\right]_{y=0}^{1-x}dx=\int_{x=0}^1\left(\frac{(1-x)^3}{3}-x(1-x)\right)dx\\ =\frac{1}{3}\int_{t=0}^1t^3 dt-\int_{x=0}^1 x(1-x)dx.$$ Can you take it from here?