# Compute Integrals

We start with setting the Feel++ environment and loading the Feel++ library.

Set the Feel++ environment with local repository
``````import feelpp.core as fppc
import sys
app = fppc.Environment(["myapp"],config=fppc.localRepository(""))``````
Results

## 1. Integral of a function

$\int_{\Omega} f(x) dx$ is the integral of the function $f$ over the domain $\Omega$. The domain $\Omega$ is a subset of $\mathbb{R}^n$ and $f$ is a function defined on $\Omega$. The domain $\Omega$ is discretized as a mesh $\mathcal{T}$ and the function $f$ may be approximated by a finite element function $\hat{f}$. The integral is approximated by the sum of the integrals over the elements of the mesh $\mathcal{T}$. The integral over each element is computed using a quadrature formula.

We first load a mesh

Load a mesh $\mathcal{T}$
``````geo=fppc.download( "github:{repo:feelpp,path:feelpp/quickstart/laplacian/cases/feelpp2d/feelpp2d.geo}", worldComm=app.worldCommPtr() )[0]
print("geo file: {}".format(geo))
mesh = fppc.load(fppc.mesh(dim=2,realdim=2), geo, 0.1)``````
Results
`geo file: /nvme0/prudhomm/github-actions/actions-runner-4/_work/book.feelpp.org/book.feelpp.org/feelppdb/downloads/feelpp2d.geo`

Then we can define functions $f$ and compute the integral of $f$ over the mesh $\mathcal{T}$.

Compute integrals
``````from feelpp.core.integrate  import integrate

i1 = integrate(range=fppc.elements(mesh),expr="sin(x+y):x:y")
i2 = integrate(range=fppc.boundaryfaces(mesh),expr="x*nx+y*ny+z*nz:x:y:z:nx:ny:nz")
i3 = integrate(range=fppc.markedelements(mesh, "marker"), expr="1")

print(f"i1 = {i1}, i2 = {i2}, i3 = {i3}")``````
Results
`i1 = [3.24282386], i2 = [41.6], i3 = [0.]`