We measured the reaction cross sections of the two-neutron halo nucleus 11Li with solid hydrogen and carbon targets at around 31 and 41 MeV/nucleon. The neutron density distribution of 11Li was deduced for the first time by the Glauber model calculation based on the optical limit approximation. The uncertainty of the matter density of 11Li was improved, compared with earlier measurements. The present root-mean-square radius of the proton distribution agrees with the previous one derived from an optical isotope shift measurement. The present root-mean-square radii reproduce theoretical calculations by the tensor optimized shell model by assuming core excitation. This consistency suggests the possibility that 9Li in 11Li is excited and the disappearance of the N=8 shell gap of 11Li is caused by correlations originating from the nucleon force, such as the tensor and the pairing.