I am struggling with the following question:
For $b>0, a\in\mathbb{R}$ let $D_{a,b}$ be the triangle in $\mathbb{R}^{2}$ with vertices $(-1,0),(1,0)$ and $(a,b)$. Calculate $\int\int_{D_{a,b}}xydA$ in dependence of $a$ and $b$.
I know that I need to use the three vertices as the general region for my integrals. I believe that I need to determine the boundaries of both $x$ and $y$. So for $y$ we would have $0\leq y\leq b$. The part where I struggle is determining this for $x$. I need to determine $x_{min}$ and $x_{max}$ such that $x_{min}\leq x\leq x_{max}$ and then I know that I simply need to compute \begin{equation} \int_{0}^{b}y\int_{x_{min}}^{x_{max}} xdxdy\end{equation} Could somebody explain to me how I should determine $x_{min}$ and $x_{max}$?
First consider the equation of the line joining $(-1,0)$ and $(a,b)$. It is given by $y=\frac{b}{a+1}(x+1) \implies x=\frac{y(a+1)-b}{b}$.
Likewise consider the line joining $(a,b)$ to $(1,0)$. The equation of this line is given by $y=\frac{b}{a-1}(x-1) \implies x=\frac{y(a-1)+b}{b}$.
The integral can be thought of as $$\iint_{D_{a,b}}xy \,dA=\int_0^by\int_{\frac{y(a+1)-b}{b}}^{\frac{y(a-1)+b}{b}}x \, dx \,dy.$$