\[ \newcommand{\euler}{\text{e}} \newcommand{\p}{\partial} \newcommand{\px}{\p x} \newcommand{\py}{\p y} \newcommand{\pz}{\p z} \newcommand{\pf}{\p f} \newcommand{\pu}{\p u} \newcommand{\pv}{\p v} \newcommand{\pl}{\p \boldsymbol{l}} \newcommand{\d}{\text{d}} \newcommand{\dt}{\d t} \newcommand{\dx}{\d x} \newcommand{\dy}{\d y} \newcommand{\dr}{\d r} \newcommand{\dv}{\d v} \newcommand{\dz}{\d z} \newcommand{\du}{\d u} \]
For the function bounded within closed area \(\Omega\), devide \(\Omega\) into arbitrary \(n\) closed area \(\{\Delta\sigma_i | 1\le i\le n\}\), and pick arbitrary point \((\xi_i, \eta_i)\in\Delta\sigma_i\), define the double integral of function \(D\) \[ \iint_D f(x, y)\d\sigma = \lim_{\lambda\rightarrow 0}\sum_{i=1}^nf(\xi_i, \eta_i)\Delta\sigma_i \] if the limit exists for all the methods of devision. The \(f(x, y)\) is called integrand, \(f(x, y)\d\delta\) is the integral expression, \(\d\sigma\) is the area element, \(D\) is the integral area. the sum \(\displaystyle \sum_{i=1}^nf(\xi_i, \eta_i)\Delta\sigma_i\) is named integral sum.
We can reduce the one triple integral into one double integral and one single integral by two ways:
\[ T = \iiint_\Omega = \int_0^d \dz \int_0^{2\pi}\d\theta \int_0^r f(r\cos\theta r\sin\theta, z)r\dr \]
\[ \left\{\begin{array}{ll} x = r\sin\varphi \cos\theta \\ y = r\sin\varphi \sin\theta \\ z = r\cos\varphi \end{array}\right. \\ \dv = \rho^2 \sin\varphi ~ \d\rho\d\theta\d\varphi \]
\[ (x^2+y^2+z^2)^2 = a^3z (a>0) \\ \rho^4 = a^3 \rho \cos\varphi \]