Properties of Double Integral
Consider two functions f(x, y) and g(x, y) to be integrated over regions A and B respectively. Also, consider C and D to be sub-regions of A and B, then a double integral satisfies the following properties:
- Linearity: [Tex]\int \int_R[f(x,y)\pm g(x,y)] = \int \int_Rf(x,y)\pm\int \int_Rg(x,y) [/Tex]
- [Tex]\int^b_a\int^d_cf(x,y)dy~dx = \int^d_c\int^b_af(x,y)dy~dx [/Tex]
- Additivity: If [Tex]R = S \cup T, S \cap T = \phi [/Tex]then, [Tex]\int\int_Rf(x,y)dy~dx = \int\int_Sf(x,y)dy~dx + \int\int_Tf(x,y)dy~dx [/Tex]
- Monotonicity: If f(x, y) ≥ g(x, y), then [Tex]\int \int_R f(x,y)dy~dx \geq \int\int_Rg(x,y)dy~dx [/Tex]
- [Tex]\int \int_R kf(x,y)dy~dx = k \int\int_R f(x,y)dy~dx [/Tex]
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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