section_path: Computation/Integral calculus
section_name: Integral calculus
paragraph_text: Integral calculus
  
  The calculation of volume is a vital part of integral calculus. One of which is calculating the volume of solids of revolution, by rotating a plane curve around a line on the same plane. The washer or disc integration method is used when integrating by an axis parallel to the axis of rotation. The general equation can be written as: V = π ∫ a b | f (x) 2 − g (x) 2 | d x {\displaystyle V=\pi \int _{a}^{b}\left|f(x)^{2}-g(x)^{2}\right|\,dx} where f (x) {\textstyle f(x)} and g (x) {\textstyle g(x)} are the plane curve boundaries. The shell integration method is used when integrating by an axis perpendicular to the axis of rotation. The equation can be written as: V = 2 π ∫ a b x | f (x) − g (x) | d x {\displaystyle V=2\pi \int _{a}^{b}x|f(x)-g(x)|\,dx} The volume of a region D in three-dimensional space is given by the triple or volume integral of the constant function f (x, y, z) = 1 {\displaystyle f(x,y,z)=1} over the region. It is usually written as: ∭ D 1 d x d y d z. {\displaystyle \iiint _{D}1\,dx\,dy\,dz.}
  
  In cylindrical coordinates, the volume integral is ∭ D r d r d θ d z, {\displaystyle \iiint _{D}r\,dr\,d\theta \,dz,}
  
  In spherical coordinates (using the convention for angles with θ {\displaystyle \theta } as the azimuth and φ {\displaystyle \varphi } measured from the polar axis; see more on conventions), the volume integral is ∭ D ρ 2 sin φ d ρ d θ d φ. {\displaystyle \iiint _{D}\rho ^{2}\sin \varphi \,d\rho \,d\theta \,d\varphi.}