Integration with parameter

Let $f: {\bf R}^n \rightarrow {\bf R}$ and an $(n+1)$-dimensional volume

$$\int_Q f(n) dV = \lim_{m \rightarrow \infty} \sum_i f( e_i ) \Delta V,$$

where $\Delta V$ a $n$-dimensional volume of an $n$-dimensional box, is the Riemann integral.

The integral of the derivative is the derivative of the integral.

Suppose $f: {\bf R}^2 \rightarrow {\bf R}$ is continuous on $Q = {\bf R} \times [ r, d ]$, and $\frac{\partial f}{\partial x}$ is continuous on $Q$. Then

$$\varphi(x) = \int_c^d f( x, y ) dy$$

is differentiable on ${\bf R} \times ( r, d )$ and

$$φ'(x) = \int_c^d \frac{\partial f}{\partial x} ( x, y ) dy,$$

Note 1: $x$ is the "parameter" here.

Note 2: also

$$φ′(x) = \left( \int_c^d f( x, y ) dy \right)'.$$

Exercise. Let

$f( x, y ) = x^2 y^3$, (try also $f( x, y ) = x^y$).

Then

$$\begin{array}{} \varphi(x) &= \int_0^1 f( x, y ) dy \\ &= ... {\rm \hspace{3pt} explain \hspace{3pt} here} \\ \varphi′(x) &= \int_0^1 f'( x, y ) dy \\ &= \int_0^1 2x y^3 dy \\ &= \left. 2x \frac{y^4}{4} \right|_0^1 = \frac{x}{2}. \end{array}$$