# volume bounded by 2d function

Let's say I have 3 points in $p_1, p_2, p_3 \in \mathbb{R}^2$ that define a triangle $T$

I define now the function $f_1 : \mathbb{R}^2 \to \mathbb{R}$ such that:

• $f_1(p_1) = 1, f_1(p_2) = f_1(p_3) = 0$
• $f_1 = 0$ outside of $T$
• $f_1$ is linearly interpolated inside $T$, given the extremities.

Likewise we define $f_2$ such that $f_2(p_2) = 1, f_2(p_1) = f_2(p_3) = 0$ and $f_3$0

What I want to compute is the integral over $\mathbb{R}^2$ (or $T$) of $f_i \cdot f_j$ where $i,j \in \{1,2,3\}$.

How would I do that? I feel like there is a simple formula to do this but all I can't manage to write it down

Thanks!

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thanks for the latexification! – lezebulon Mar 21 '13 at 18:16
Are you familiar with finite elements? – Occupy Gezi Mar 21 '13 at 19:39
@Anil Baseski : I'm trying to get this formula to use it to compute finite elements – lezebulon Mar 21 '13 at 19:51
Check this mems.rice.edu/~akin/Elsevier/Chap_10.pdf – Occupy Gezi Mar 21 '13 at 20:16