Double integral with triangular region of integration
I'm attempting to do a Calc III homework problem, and I feel like I'm on the right track, but somewhere I either mess up or set up the problem incorrectly and I don't know how or why.
Here is the problem $$ \text{Evaluate the double integral }\int \int_{D}^{}xy\text{ }dA\\ \text{ where } D \text{ is the triangular region with vertices }(0,0)\text{, }(6,0)\text{, }(0,5)\text{.} $$
This seems like it should be straight-forward. I drew a picture of the vertices, and created the triangle. Then I decided that y axis the region could be described as $\frac{-5}{6}x+5$ the integral can be written as $$ \int_{0}^{6}\int_{0}^{\frac{-5}{6}x+5}xy\text{ }dy dx $$ So solving: $$ \frac{1}{2}\int_{0}^{6} [xy^2]|_{0}^{\frac{-5}{6}x+5}\text{ }dx\\ \frac{1}{2}\int_{0}^{6} x(\frac{-5}{6}x+5)^2dx\\ \frac{25}{2}\int_{0}^{6}\frac{x^2}{36}-\frac{2x}{6}+1dx\\ \frac{25}{2}[\frac{x^3}{108}-\frac{2x^2}{12}+x]_{0}^{6}\\ \frac{25}{2}(2 - 6 +6) = 25 $$
It seems straight forwards, which leads me to think I setup my region incorrectly. Can some tell me what I did wrong?
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