  The matrix integral has many applications in diverse fields. This review
article begins by presenting detailed key background knowledge about matrix
integral. Then the volumes of orthogonal groups and unitary groups are
computed, respectively. As a unification, we present Mcdonald's volume formula
for a compact Lie group. With this volume formula, one can easily derives the
volumes of orthogonal groups and unitary groups. Applications are also
presented as well. Specifically, The volume of the set of mixed quantum states
is computed by using the volume of unitary group. The volume of a metric ball
in unitary group is also computed as well.
  There are no new results in this article, but only detailed and elementary
proofs of existing results. The purpose of the article is pedagogical, and to
collect in one place many, if not all, of the quantum information applications
of the volumes of orthogonal and unitary groups.
