Why is the integration formula in localization called the integration formula?
Let \(M\) be a compact, oriented real manifold of dimension \(n\). Then we have a pairing between de Rham cohomology and singular cohomology \[\langle - , - \rangle: H^k_{dR}(M) \times H_k(M;\mathbb R)\to \mathbb R\] which is given by integrating a closed differential \(k\)-form \(\omega\) representing a class in \(H^k_{dR}(M)\) along a dimension \(k\) submanifold \(Z\) representing a class \([Z]\in H_k(M;\mathbb R)\):
\[\langle[\omega], [Z]\rangle = \int_{Z}\omega\] and then extending linearly. Indeed, this is well defined. Since \(\omega\) is closed (\(d\omega = 0\)) we can modify \(\omega\) by adding an exact form \(d\eta\) without changing the integral by Stoke’s theorem: \[\int_Z \omega + d\eta = \int_Z \omega + \int_{dZ}\eta = \int_Z\omega\] where the last equality follows from the fact that \(Z\) is a closed submanifold; i.e. it has no boundary since it represents a homology class and thus \(\partial Z = 0\).
Now, given the map \(\rho:M\to \pt\), we can take the Gysin pushforward to obtain a map \[\rho_*:H^k(M)\to H^{k - (\dim M)}(\pt)\] which is zero unless \(k = \dim (M) = n\), in which case we get \[\rho_*:H^n(M)\to \mathbb R.\] It turns out this is exactly the pairing above, i.e. \[\rho_*([\omega]) = \langle [\omega], [M]\rangle = \int_M\omega.\] We call the pushforward/integration formula in equivariant cohomology the integration formula by analogy, but in scenarios where you have a surjection \(H^*_T(X) \to H^*(X)\), you can actually compute \(\int_M\omega\) using the localization toolkit.