Calculus

Leibniz Integral Rule

When can we swap the order of integration and differentiation? Suppose we have

\[\partial_{y} \int_{a(y)}^{b(y)} f(x, y) dx\]

Using the Fundamental Theorem of Calculus, we can answer this question. First, replace the integral with the integrated function:

\[\begin{align*} \partial_{y} \int_{a(y)}^{b(y)} f(x, y) dx &=\partial_{y} [F(b(y), y) - F(a(y), y)]\\ &= f(b(y), y) \partial_y b(y) + f(b(y), y) - f(a(y), y) \partial_y a(y) + f(a(y), y)\\ &= f(b(y), y) \partial_y b(y) - f(a(y), y) \partial_y a(y) + \int_{a(y)}^{b(y)} \partial_y f(x, y) dx\\ \end{align*}\]

This is also commonly called differentiating under the integral sign.