Integral depending on a parameter

This note collects two standard “Leibniz-type” rules from Riemann integration.

Variable limits: $φ (x) = \int_{a (x)}^{b (x)} f (t) d t$

Assume $f$ is Riemann integrable on an interval $I$ and has an antiderivative $F$ on $I$ (e.g. it is continuous). Let $a, b : D \to I$ .

If $a$ and $b$ are continuous, then

φ (x) = \int_{a (x)}^{b (x)} f (t) d t

is continuous on $D$ .

If $a$ and $b$ are differentiable, then $φ$ is differentiable and

φ^{'} (x) = f (b (x)) b^{'} (x) - f (a (x)) a^{'} (x) .

Parameter in the integrand: $φ (x) = \int_{a}^{b} f (x, t) d t$

Let $U \subset R$ be open and $f : U \times [a, b] \to R$ .

If $f$ is continuous, then

φ (x) = \int_{a}^{b} f (x, t) d t

is continuous on $U$ .

If additionally $\partial f / \partial x$ exists and is continuous on $U \times [a, b]$ , then $φ$ is differentiable and

φ^{'} (x) = \int_{a}^{b} \frac{\partial f}{\partial x} (x, t) d t .