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An introduction to Double Integral: Its Basics and
Examples

Leibniz and Newton introduced the fundamental idea of the integral in the 17th century, but Barrow and
Torricelli laid the groundwork for the relationship between integration and derivation. Gottfried Wilhelm
Leibniz initially introduced the sign for integration in 1645; it is the opposite of derivation.
We will examine integration and its various forms in this article. Basic integration formulae or rules, and an
overview.

What is double integral?

A double integral is a mathematical tool used to calculate the area or volume of a two-dimensional or
three-dimensional region.
The double integral is denoted by ∬, and it represents the sum of infinitesimal areas or volumes over a
two-dimensional or three-dimensional region, respectively.
The general definition of a double integral is as follows:
∬ f (x, y) dA
Where
f (x, y) is a function of two variables
dA represents an infinitesimal area element.
The integral is taken over a region R in the xy-plane, and the limits of integration are determined by the
bounds of the region.

Double integral Rule:

In calculus, we typically perform any integration approach by following the rules and formulas. You must
have studied a variety of strategies, including parts integration, substitution integration, and formula
integration, in order to answer integration problems.
We shall explore the rule for double integration by parts, which is given by, in this instance as well as the
case of double integration.
∫∫u dv / dx dx.dy = ∫[uv -∫v du/dx dx]dy
Characteristics of double integral:
 ∫ a b  ∫ c d  f(x, y) dy.dx = ∫ c d ∫ a b  f(x, y) dx.dy
 ∫∫ (f(x, y) ± g(x, y)) dA = ∫∫ f(x, y) dA ± ∫∫g(x, y) dA
 If f(x, y) < g(x, y), then ∫∫ f(x, y) dA < ∫∫ g(x, y) dA
 k ∫∫ f(x, y) . dA = ∫∫ k . f(x, y) . dA
 ∫∫ R∪S f(x, y) . dA = ∫∫ R f(x, y) . dA + ∫∫ s f(x, y) . dA
Aera of double integral:
We need to determine the double integral of z, where z = f(x, y) is specified across a domain D in the x-y
plane. The Double Integral Formula can be used if the necessary region is divided into vertical stripes and
the region's bounds, or the endpoints for x and y, are precisely determined:
 ∫∫ D f(x, y) dA = ∫ x=a x=b  ∫ y=f1(x) y=f2(x)  f(x, y) dy.dx
Additionally, if we carefully determine the x and y endpoints, or the region's boundaries, and divide the
necessary region into horizontal stripes, we can use the following formula:
 ∫∫ D f(x, y) dA = ∫ y=c y=d  ∫ x=g1(y) x=g2(y)  f(x, y) dx.dy
For continuous function:
 ∫∫ D f(x, y) dA = ∫ x=a x=b  ∫ y=f1(x) y=f2(x)  f(x, y) dy.dx = ∫ y=c y=d  ∫ x=g1(y) x=g2(y)  f(x, y) dx.dy
Polar coordinates of double integral:
Doubling integral
∫∫ R f(x, y) dA
A double integral in polar coordinates can be created from n rectangle coordinates as follows:
∫∫ R f(x, y) dA = ∫∫ R f(r cos��, r sin��) dr d��

Another way is:
∫∫ R f(x, y) dA = ∫ ��1 ��2  ∫ r1 r2  f(r cos��, r sin��) dr d��
Here,
f(r cos��, r sin��) = f(r, ��)
By treating it as a constant, we must first integrate f(r, ��) with respect to r within the limits of r = r 1 and r =
r 2 , and then we must integrate the resulting equation with respect to �� from �� 1 to �� 2 . In this case, r 1 and
r 2 could be constants or functions of ��.
In this instance, we must first integrate f(r, ��) with respect to �� between the limits of �� = �� 1 and �� =
�� 2 , while treating r as a constant. Once the resulting expression is integrated wrt r, the function of �� will
be constant.
Application of double integral:
Here are a few examples of how double integrals are used in daily life:
 Calculating volumes and areas: Double integrals are commonly used to calculate the volumes and
areas of complex shapes.
 Heat and mass transfer: Double integrals are used to calculate the rate of heat and mass transfer in
various systems, such as in heat exchangers or fluid flow analysis.
 Probability and statistics: Double integrals are used to calculate joint probability density functions
and cumulative distribution functions, which are important in statistical analysis.
 Electrical engineering: Double integrals are used to calculate the total energy stored in a three-
dimensional electrical field.
 Computer graphics: Double integrals are used in computer graphics to calculate the illumination of
a three-dimensional scene.
Overall, double integrals are a powerful mathematical tool with numerous practical applications in many
fields of science, engineering, and technology.
How to calculate double integral?
Taking assistance from online tools is an easy way to find the steps and results of the double integral calculator. You
can use a double integral calculator in order to save time. Below are a few examples to solve the problems of double
integral manually.
Example 1:
Let's say we want to find the volume of a solid region in three-dimensional space that is bounded by the
planes x = 0, y = 0, z = 0, x + y + z = 1. We can find this volume by evaluating a triple integral, but here we
will use a double integral to simplify the calculation.
Solution:
Step 1: We see that the region is a triangle with vertices at (0,0), (1,0), and (0,1). So, the limits of
integration for the double integral will be:
0 ≤ x ≤ 1-y
0 ≤ y ≤ 1-x
Step 2: Now, we need to determine the integrand. Since we want to find the volume of the solid, we can
integrate the constant function f(x, y, z) = 1 over the region R. Thus, the double integral is:

∬ R 1 dA
∬ R 1 dA = ∫₀¹ ∫₀¹-x 1 dy dx
∬ R 1 dA = ∫₀¹ ∫₀¹-x dy dx
∬ R 1 dA = ∫₀¹ (1-x) dx
∬ R 1 dA = ½
So, the volume of the solid is ½ cubic unit.
Example 2:
Suppose we want to calculate the area of the region R in the xy-plane bounded by the curves y = x and y =
x 2
Solution:
Step 1: We can set up the double integral as follows:
∬ R 1 dA
Step 2: The region R can be described by the bounds 0 ≤ x ≤ 1 and x 2 ≤ y ≤ x. So, the limits of integration for
the double integral are:
∬ R 1 dA = ∫ 0 1 ∫ x^2 x 1 dy dx
Step 3: Evaluating the inner integral first, we get:
∫ 0 1 ∫ x^2 x 1 dy dx = ∫ 0 1 x – x 2 dx
Step 4: Integrating with respect to x, we get:
∫ 0 1 ∫ x^2 x 1 dy dx = [x 2 / 2 – x 3 / 3] 0 1
∫ 0 1 ∫ x^2 x 1 dy dx = (1/2 – 1/3) – (0 – 0)
∫ 0 1 ∫ x^2 x 1 dy dx = 1/6
So, the area of region R is 1/6 square units.
Conclusion
In this article, we studied the history of integral and what is double integral, basic rules of a double
integral. Also discussed are the characteristics of a double integral. In this article, we also discussed how to
calculate the area using a double integral. We discussed the polar coordinates of a double integral.

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