MHB What is the correct way to apply Green's Theorem in this scenario?

[mathmari]

Hey!

I want to compute the integral $\oint_C \cos \left (x^{2017}\right )dx+\left (\frac{x^2}{2}+\sin y^{2018}\right )dy$, where $C$ is the boundary of the bounded field that is defined by the curves $y=2-x^2$ and $y=x$, with positive orientation.

We have to apply Green's Theorem, or not?

So, we get the following: $$\oint_C \cos \left (x^{2017}\right )dx+\left (\frac{x^2}{2}+\sin y^{2018}\right )dy=\iint_D \left ( \frac{\partial}{\partial{x}}\left[\frac{x^2}{2}+\sin y^{2018}\right ]-\frac{\partial}{\partial{y}}\left [\cos \left (x^{2017}\right )\right ]\right )dxdy=\iint_D 2xdxdy$$ where $D=\{-2\leq x\leq 1, \ 2-x^2\leq y\leq x\}$.

Therefore, we get: $$\iint_D 2xdxdy=\int_{-2}^1\int_{2-x^2}^x 2xdydx=\frac{9}{2}$$

Is this correct? (Wondering)

 

[Opalg]

Hey!

I want to compute the integral $\oint_C \cos \left (x^{2017}\right )dx+\left (\frac{x^2}{2}+\sin y^{2018}\right )dy$, where $C$ is the boundary of the bounded field that is defined by the curves $y=2-x^2$ and $y=x$, with positive orientation.

We have to apply Green's Theorem, or not?

So, we get the following: $$\oint_C \cos \left (x^{2017}\right )dx+\left (\frac{x^2}{2}+\sin y^{2018}\right )dy=\iint_D \left ( \frac{\partial}{\partial{x}}\left[\frac{x^2}{2}+\sin y^{2018}\right ]-\frac{\partial}{\partial{y}}\left [\cos \left (x^{2017}\right )\right ]\right )dxdy=\iint_D 2xdxdy$$ where $D=\{-2\leq x\leq 1, \ 2-x^2\leq y\leq x\}$.

Therefore, we get: $$\iint_D 2xdxdy=\int_{-2}^1\int_{2-x^2}^x 2xdydx=\frac{9}{2}$$

Is this correct? (Wondering)

Looks right to me. (Yes)

 

[mathmari]

Looks right to me. (Yes)

Great! Thank you very much! (Happy)

 

[Opalg]

Great! Thank you very much! (Happy)

Actually no, it doesn't look quite right. Shouldn't the partial derivative be $x$ rather than $2x$?

 

[mathmari]

Actually no, it doesn't look quite right. Shouldn't the partial derivative be $x$ rather than $2x$?

Oh yes, you're right! (Tmi)