Using Tonelli - upper and lower limits of the integral

Question

Given $$f(x,y)=x^2+y^2$$ and a triangle $A$ with the corner points $$(0,0),(0,1),(\frac{1}{2},\frac{1}{2})$$ how do I compute $$\int_A fd\lambda^2$$ I know I have to use Tonelli, but do I have to split the triangles into two triangles or how can I use Tonelli straightforward? Some help and explanation would be appreciated, especially when it comes to finding the lower and upper limits of the integral..

Answer 1

Since $A$ is compact and $f$ is continuous, there exists $(x^\star,y^\star)\in A$ with $$M:=f(x^\star,y^\star)=\sup_{(x,y)\in A}f(x,y).$$ $f$ is nonnegative, so we have $$\int_A f\ \mathsf d\lambda^2\leqslant M\lambda^2(A)<\infty, $$ and hence $f\in L^2(A)$. The assumptions of Tonelli's theorem are satisfied, so we may compute the double integral as an iterated integral: \begin{align} \int_A f\ \mathsf d\lambda^2 &= \iint\limits_A f(x,y)\ \mathsf dx\ \mathsf dy\\ &= \int_0^{\frac12}\int_0^y(x^2+y^2)\mathsf dx\ \mathsf dy+\int_{\frac12}^1\int_0^{1-y}(x^2+y^2)\mathsf dx\ \mathsf dy\\ &= \frac1{48} + \frac1{16}\\ &= \frac1{12}. \end{align}