diff --git a/docs/1b/math119.md b/docs/1b/math119.md index caf6c43..4c8cf1c 100644 --- a/docs/1b/math119.md +++ b/docs/1b/math119.md @@ -561,3 +561,31 @@ $$\iint_R f(x,y)dA$$ &=\frac 2 3u^{3/2}du\biggr|^1_0 \\ &=\frac 2 3\pi \end{align*} + +## Triple integration + +Much like double integrals: + +The **volume** within bounds $E$ is the integral of 1: + +$$V=\iiint_E1dV$$ + +The **average value** within a volume is: + +$$\overline f_E=\frac 1 V\iiint_Ef(x,y,z)dV$$ + +!!! example + For the volume within $x+y+z=1$ and $2x+2y+z=2,x,y,z\geq 0$: + + The points intersect the axes and each other to create the bounds $0\leq x\leq 1,0\leq y\leq 1-x,1-x-y\leq z\leq 2-2x-2y$. + + $$\int^1_0\int^{1-x}_0\int^{2-2x-2y}_{1-x-y}1dz\ dy\ dx =\frac 1 6$$ + + The average value is: + + $$6\iiint_Ez\ dV=\frac 3 4$$ + +The **total quantity** if $f$ represents density is: + +$$T=\iiint_Ef(x,y,z)dV$$ +