section_path: In calculus/Line, surface, and volume integrals
section_name: Line, surface, and volume integrals
paragraph_text: Line, surface, and volume integrals
  
  For some scalar field f : U ⊆ R → R, the line integral along a piecewise smooth curve C ⊂ U is defined as
  
  where r: → C is an arbitrary bijective parametrization of the curve C such that r (a) and r (b) give the endpoints of C and a < b {\displaystyle a<b}.
  
  For a vector field F : U ⊆ R → R, the line integral along a piecewise smooth curve C ⊂ U, in the direction of r, is defined as
  
  where · is the dot product and r: → C is a bijective parametrization of the curve C such that r (a) and r (b) give the endpoints of C.
  
  A surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analog of the line integral. To find an explicit formula for the surface integral, we need to parameterize the surface of interest, S, by considering a system of curvilinear coordinates on S, like the latitude and longitude on a sphere. Let such a parameterization be x (s, t), where (s, t) varies in some region T in the plane. Then, the surface integral is given by
  
  where the expression between bars on the right-hand side is the magnitude of the cross product of the partial derivatives of x (s, t), and is known as the surface element. Given a vector field v on S, that is a function that assigns to each x in S a vector v (x), the surface integral can be defined component-wise according to the definition of the surface integral of a scalar field; the result is a vector.
  
  A volume integral is an integral over a three-dimensional domain or region.When the integrand is trivial (unity), the volume integral is simply the region's volume. It can also mean a triple integral within a region D in R of a function f (x, y, z), {\displaystyle f(x,y,z),} and is usually written as: