The ith mean curvature \(K_ i\) of a compact immersed submanifold of dimension n in \(E^ k\) is the normalized ith elementary symmetric function of the principal curvatures. The authors consider homothety- invariant integrals of functions of the \(K_ i\). They discuss lower bounds for these. A major result states that for each i, n, and positive \(\epsilon\) there exists a hypersurface with arbitrary first Betti number, and \[ 2\leq (1/c_ n)\leq \int | K_ i|^{n/i} dV\leq 4+\epsilon. \] Some estimates for the inf of the Willmore functional are deduced.