  We introduce the notions of categorical integrals and categorical cointegrals
of a finite tensor category $\mathcal{C}$ by using a certain adjunction between
$\mathcal{C}$ and its Drinfeld center $\mathcal{Z}(\mathcal{C})$. These notions
can be identified with integrals and cointegrals of a finite-dimensional Hopf
algebra $H$ if $\mathcal{C}$ is the representation category of $H$. We
generalize basic results on integrals and cointegrals of a finite-dimensional
Hopf algebra (such as the existence, the uniqueness, and the Maschke theorem)
to finite tensor categories. Motivated by results of Lorenz, we also
investigate relations between categorical integrals and morphisms factoring
through projective objects. Finally, we extend the $n$-th indicator of a
finite-dimensional Hopf algebra introduced by Kashina, Montgomery and Ng to
finite tensor categories.
