The Wolfram Language deeply integrates regions into high-level solvers, including the ability to integrate or solve partial differential equations over regions, solve equations and inequalities with region constraints, or optimize over regions. The results can be either symbolic and exact or numeric and approximate.

Integrals

Integrate, NIntegrate — integrate over a region

Probability  ▪  NProbability  ▪  Expectation  ▪  NExpectation

Partial Differential Equations »

NDSolve — solve a partial differential equation over a region

NDEigensystem — compute eigenvalues and eigenfunctions over a region

NDSolveValue  ▪  NDEigenvalues  ▪  DirichletCondition  ▪  NeumannValue  ▪  ...

DSolve  ▪  DSolveValue  ▪  DEigensystem  ▪  DEigenvalues  ▪  ...

Equations & Inequalities

Solve — solve equations, inequalities, and region constraints

NSolve  ▪  Reduce  ▪  FindInstance  ▪  Resolve  ▪  SolveValues  ▪  NSolveValues

Optimization »

Minimize — minimize a function over a region

MinValue  ▪  ArgMin  ▪  NMinimize  ▪  NMinValue  ▪  NArgMin  ▪  FindMinimum  ▪  FindMinValue  ▪  FindArgMin  ▪  Maximize  ▪  MaxValue  ▪  ArgMax  ▪  ...

Visualization »

Plot3D — plot a function over a region

ContourPlot  ▪  DensityPlot  ▪  VectorPlot  ▪  StreamPlot  ▪  ...

SliceContourPlot3D  ▪  SliceDensityPlot3D  ▪  SliceVectorPlot3D  ▪  DensityPlot3D  ▪  ...

Geometric Solvers »

RegionNearest — find the nearest point in a region

RegionMeasure  ▪  RegionCentroid  ▪  RegionDistance  ▪  ...

Synthetic Geometry »

GeometricScene  ▪  RandomInstance  ▪  FindGeometricConjectures  ▪  ...