So I decided to evaluate some double and triple integrals for fun. After a while, I came up with a sort of difficult triple integral (overall out of the integrals that I created during my session):$$\int_Df(x,y,z)dxdydz$$where$$f(x,y,z)=\dfrac{x+y+z}{xyz},D=\left\{(x,y,z)\in\mathbb R^3,0.01\le x\le e,0.01\le y\le e,0.01\le z\le e\right\}$$This I thought that I might be able to integrate. Here is my attempt at doing so:$$\begin{align}\int_Df(x,y,z)dxdydz=&\iiint_{[0.01,e]^3}\dfrac{x+y+z}{xyz}dxdydz\\\implies&\iiint_{[0.01,e]^3}\dfrac1{yz}+\dfrac1{xz}+\dfrac1{xy}dxdydz\\\implies&\iint_{[0.01,e]^2}\dfrac{e-0.01}{yz}+\ln\left(100e\right)\left(\dfrac1z+\dfrac1y\right)dydz\\\implies&\iint_{[0.01,e]^2}\dfrac1y\left(\dfrac{e-0.01+\ln(100e)}z\right)dydz\end{align}$$which can be rewritten as$$\begin{align}(e-0.01+\ln(100e))\int_{0.01}^e\dfrac{dy}y\int_{0.01}^e\dfrac{dz}z=&(e-0.01+\ln(100e))\ln^2(100e)\\=&e\ln^2(100e)-0.01\ln^2(100e)+\ln^3(100e)\\\approx&261.191476841396743068954331\end{align}$$as given by Wolfram Alpha.
My question
Did I evaluate the integral correctly, or what could I do to evaluate it over the domain that is given?
Note: I usually only ask if I evaluated an integral correctly if I had a lot of trouble integrating it and am unsure about my solution.
Best Answer
Assuming $0<a<b$:
$$\begin{align} &\iiint_{[a,b]^3}\dfrac{x+y+z}{xyz}dxdydz=\iiint_{[a,b]^3}\left(\dfrac1{yz}+\dfrac1{xz}+\dfrac1{xy}\right)dxdydz\\ \\ &=\int_a^bdx\int_a^b\frac{dy}y\int_a^b\frac{dz}z+ \int_a^b\frac{dx}x\int_a^bdy\int_a^b\frac{dz}z+ \int_a^b\frac{dx}x\int_a^b\frac{dy}y\int_a^bdz\\ \\ &=3(b-a)\ln^2\frac ba. \end{align}$$