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Just as the definite integral of a positive function of one variable represents the area of the region between the graph of the function and the x-axis, the double integral of a positive function of two variables represents the volume of the region between the surface defined by the function (on the three-dimensional Cartesian plane where z = f(x, y)) and the plane which contains its domain. [1]
This article uses the standard notation ISO 80000-2, which supersedes ISO 31-11, for spherical coordinates (other sources may reverse the definitions of θ and φ): . The polar angle is denoted by [,]: it is the angle between the z-axis and the radial vector connecting the origin to the point in question.
It can also mean a triple integral within a region of a function (,,), and is usually written as: (,,).. A volume integral in cylindrical coordinates is (,,), and a volume integral in spherical coordinates (using the ISO convention for angles with as the azimuth and measured from the polar axis (see more on conventions)) has the form (,,) .
To find an explicit formula for the surface integral of f over S, we need to parameterize S by defining a system of curvilinear coordinates on S, like the latitude and longitude on a sphere. Let such a parameterization be r(s, t), where (s, t) varies in some region T in the plane. Then, the surface integral is given by
Shell integration (the shell method in integral calculus) is a method for calculating the volume of a solid of revolution, when integrating along an axis perpendicular to the axis of revolution. This is in contrast to disc integration which integrates along the axis parallel to the axis of revolution.
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 ...
The first two integrals are iterated integrals with respect to two measures, respectively, and the third is an integral with respect to the product measure. The partial integrals ∫ Y f ( x , y ) d y {\textstyle \int _{Y}f(x,y)\,{\text{d}}y} and ∫ X f ( x , y ) d x {\textstyle \int _{X}f(x,y)\,{\text{d}}x} need not be defined everywhere, but ...
Cylindrical coordinate system § Line and volume elements; Measure (mathematics) – Generalization of mass, length, area and volume; Poincaré metric provides a review of the volume form on the complex plane; Spherical coordinate system § Integration and differentiation in spherical coordinates