Help for evaluating the line integrals with Green's Theorem

Question

Earlier, when I scrolled the Instagram posts I found a mathematical problem uploaded by The Vegan Math Guy like the following because this problem looks interesting to me to be solved. The mathematical problem is to find $\displaystyle \int\limits_C x^4 \,\mathrm dx+ xy \,\mathrm dy$ where $C$ is the area shown by the following graph.

In my solution, I'd use Green's Theorem because I've seen this line same as told in Ron Larson's calculus book at Theorem 15.8. Green's Theorem says that:

Let $R$ be a simply connected region with a piecewise smooth boundary $C$, oriented counterclockwise (that is $C$ is traversed once so that the region $R$ always lies to the left). If $M$ and $N$ have continuous first partial derivatives in an open region containing $D$, then $$\displaystyle \int\limits_C M\,\mathrm dx+N\,\mathrm dy=\iint\limits_R\left( \frac{\partial N}{\partial x}-\frac{\partial M}{\partial y} \right)\,\mathrm dA$$

Then I follow the steps as demonstrated in Example 1 on p. 1025, like this:

• Examine the conditions
  1. Is $C$ positively oriented? Yes.
  2. Is $C$ piecewise smooth? Yes.
  3. Is $C$ closed and simply connected? Yes.
• Find the partial derivative the function have continuous first partial derivatives.

Let $M=xy$ and $N=x^4$. \begin{align*} \frac{\partial M}{\partial y}\implies\frac{\partial }{\partial y}\left[ xy \right]&=x\frac{\partial}{\partial y}[y]\\&=x \\ \frac{\partial N}{\partial x}\implies\frac{\partial }{\partial x}\left[ x^4 \right]&=4x^3 \end{align*}

• Then taking Green's Theorem, I evaluated like this: \begin{align*} \displaystyle \int\limits_C x^4\,\mathrm dx+xy\,\mathrm dy&=\iint\limits_D\left( \frac{\partial N}{\partial x}-\frac{\partial M}{\partial y} \right)\,\mathrm dA \\&=\int\limits_0^1\int\limits_{0}^{1-x}\left( 4x^3 \right)-x\,\mathrm dy\mathrm dx\\&= \int\limits_0^1 \left[ 4x^3y-xy \right]^{1-x}_0\,\mathrm dx\\&=\int\limits_0^1\left( 4x^3(1-x)-x(1-x) \right)\,\mathrm dx\\&=\int\limits_0^1 4x^3-4x^4-x+x^2\,\mathrm dx\\&=\left[ x^4-\frac{4x^5}{5}-\frac{x^2}{2}+\frac{x^3}{3} \right]_0^1\\&=1-\frac{4}{5}-\frac12+\frac13\\&=\dfrac{1}{30} \end{align*}

After evaluating the line integral, I see why my line integral is different than the one of the comments on Instagram @a_generic_nerd2 that shares his solution. This user got $\frac16$ while me $\frac{1}{30}$, because I making that taking the partial derivative of $x^4$ with respect to $x$ for $N$ and for $M$, I taking the partial derivative of $xy$ with respect to $y$, but this user does reverse from what I do on the partial derivative. In your opinion, am I doing wrong?

Here's the link that leads to the source I mentioned in this problem.