We introduce a novel method to compute approximations of contour integrals. The new method is based on the coarea formula in combination with a Galerkin projection. As such it fits seamlessly into the spirit of hp/spectral finite element methods and circumvents the expensive and technical computation of contours. We provide convergence estimates showing that high order convergence can be achieved provided the data is sufficiently smooth. The theoretical results are supplemented by extensive numerical experiments for an example application from plasma modeling in nuclear fusion.