# Differentiating General Integrals #

*by Shuang Zhao*

In the previous section, we have discussed the differentiation of 1D Riemann integrals.

In what follows, we discuss the differentiation of a general Lebesgue integral `$I(\theta)$`

over some bounded and open domain `$\Omega$`

associated with measure `$\mu$`

:

\begin{equation}
\label{eqn:I}
I(\theta) = \int_{\Omega} f(\bx, \,\theta) \,\D\mu(\bx).
\end{equation}

For practical rendering problems, the domain `$\Omega$`

can be:

- The surface of the unit sphere
`$\sph := \{ \bx \in \real^3 : \| \bx \| = 1 \}$`

; - The union
`$\calM$`

of all object surfaces; - The path space (under Veach’s path-integral formulation).

Assuming that `$f(\bx, \theta)$`

is piecewise continuous with a zero-measure extended boundary `$\overline{\partial\Omega}$`

comprising the domain boundary `$\partial\Omega$`

and all jump discontinuity points denoted as `$\Delta\Omega$`

:

Although this assumption may seem restrictive, it is the case for most, if not all, problems in rendering.

## Reynolds Transport Theorem #

TBD.

## Computing Boundary Change Rates #

TBD.