Abstract: A good way to put two subspaces or two matrices into general position is to randomly rotate one by an orthogonal or unitary matrix, randomly selected according to Haar measure. Then in order to find expected values, one has to compute matrix integrals. Adolf Hurwitz gave a construction of Haar measure on the $N \times N$ unitary group in 1897; many now consider his paper to be the beginning of random matrix theory. In 1978, Don Weingarten gave a new method based on Schur-Weil duality; Weingarten's method is now known as the \textit{Weingarten calculus}. While exact formulas are possible, they can be quite complicated. If one is willing to settle for an approximate answer, there is an expansion in $1/N$ which is much simpler. The Weingarten function has become central to free probability. I will explain how an extension of free probability, called infinitesimal freeness, gives new insight on the Weingarten calculus.