Rules of Double Integration
The rules of double integration are mentioned below:
- The double integral is a linear operator, meaning that it satisfies the following properties: [Tex]\int \int_R[f(x,y)\pm g(x,y)] = \int \int_Rf(x,y)\pm\int \int_Rg(x,y)[/Tex] and [Tex]\int \int k.f(x, y)dA = k.\int \int f(x, y)dA[/Tex]
- The order of integration can be interchanged under certain conditions. This is known as Fubini’s theorem. For a continuous function f(x, y) defined over a rectangular region R = [a, b] × . According to Fubini’s Theorem [Tex]\int\int_R f(x, y)dA = \int_a^b\int_c^df(x, y)dydx = \int_c^d\int_a^bf(x, y)dxdy[/Tex]
- Double integrals can be evaluated using a change of variables to transform the region of integration.
- Integration by parts can be applied to double integrals to simplify the computation, especially when dealing with products of functions.
Double Integral
A double integral is a mathematical tool for computing the integral of a function of two variables across a two-dimensional region on the xy plane. It expands the concept of a single integral by integrating the functions of two variables over regions, surfaces, or areas in the plane. In case two variables are present, we need to substitute the value of one variable in terms of the other. This technique becomes very difficult when we deal with multiple variables to calculate the areas and volumes under the curves. A double integral is very useful in such cases. In this article, we will learn about double integrals in detail.
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