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Double Integral

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Double Integral is primarily used to integrate the area of a surface of a two-dimensional figure, such as rectangle, circle, square, triangle, quadrilateral, and pentagon.  ‘ ∫∫’ is known to be the mathematical symbol of double integral. Moreover, simple integration is the basis of double integral. Thus, it is crucial to understand the basics with simple integration examples before moving to a more complex double integral method. Also, double integrals assist you in computing the volume of a surface as well. It is a vital element of calculus. There are multiple types of integration in mathematics, simple integration, double integration, and triple integration. You need to understand the rules of every kind of integration to understand the other two types of integrals. 

The Geometrical Interpretation Of Double Integral

Double integral or double integration method can be defined as the process in which two variables x and y are involved, and you need to integrate with both of them. This process of integration is employed to assess the volume of a given function under a curve. Well, the geometrical interpretation of double integral can be easily defined as ‘3D volume under the surface.’ Here is the geometrical representation of an arbitrary surface when represented by a function z = f(x,y)


Image will be added soon

(The image shows the geometrical representation of double integral)

You need to divide the said domains into small rectangles as per the image.

Where 

\[\iint_{D}^{}f(x,y)IA = \sum \triangle V\] Which is the double integral


                   = \[\sum F(x,y)\triangle A\] which implies under the graph

                     

When you use double integral over the specific region or domain, you are actually adding different elements possessing a height of z and a base area of dA=dx.dy. Remember, you will have to add up these elements infinitely. This is the double integration formula.

The Main Difference Between Simple Integral and Double Integral

As mentioned above, it is imperative to understand the basic difference between the types of integrals to have a clear understanding of the concept. Whereas the simple integration helps you in finding the area under a curve, the double integration method assists you in seeking the volume under the surface. Now, this is known to be the fundamental difference between the two integrals. Technically, the simple integration of a positive function of a single variable represents the area of the specified region between the x-axis and the function. On the other hand, the double integral of two distinct variables constitutes the volume of a particular said region as between the surface as defined by the function as well as the plane containing the domain. 

Purpose Of Using Double Integrals

The basic purpose of the double integral in mathematics is to basically integrate three-dimensional functions. When in school, you would learn about a particular plane being described by x and y as well the lines and the curves that exist within these points. Double integrals are basically used for solving multiple single integrals that cannot be evaluated individually.

The Major Properties Of Double Integral

Different properties of the Double integral are parallel to those of single integrals:

  • For f and g being continuous in the region D along with C as a rational number:


             ∫∫D(f + g) dA = ∫∫D f dA + ∫∫D g dA

       

             ∫∫D cf dA = c ∫∫D f dA 


  • For f being continuous in the said region D, where D= D1 ∪ D2, where D1 and D2 are actually non-overlapping regions whose union is D: 

             ∫∫D f dA = ∫∫D1 f dA + ∫∫D2 f dA


Solved Examples

Example 1

Make use of a simple sketch to set up the double integral for the mass of the planar region represented by R possessing a variable density of δ(x, y). 

Solution: The sketch would look like:

Image to be added soon 

Area = ΔA

You need to understand that since the said region is a planar region, its density would be measured in the mass/unit area. Now, the density over a small piece of the region would not vary much, and we assume it to be continuous. Thus, if we consider that Δm is the mass of the piece, we get:  

  Δm ≈ δ(x, y)ΔA,

Here, (x, y) is known as the point in the piece, while ΔA is the piece’s area. Now, if we make slices of the region into minute pieces and calculate the masses of the given pieces, then we need to remember that the region’s total mass would be the sum of all the masses of the pieces together.

If we assume ΔA ----> 0, then the approximation will improve, and you will get the sum as an integral.

M = δ(x, y) dA.

Now, if we cut the region into  n pieces and label them from 1 to n, and Δmi is known to be the mass of the ith piece, we get:

Δmi ≈ δ(xi,yi)ΔAi,

When we sum the masses of all these pieces, we get:

Mass of R =Δmi ≈ δ(xi,yi)ΔAi. i=1


Example 2: 

Solve: where f (x, y) = \[1\sqrt{x^{2}+y^{2}}\]

\[\sqrt{x^{2}+y^{}}\iint_{R}^{}f(x,y)dxdy\;where\; f(x,y)\]= 1 / \[\sqrt{x^{2}+y^{2}}\]

And R would be the region inside the circle of radius 1. The circle would be centered at (1,0)


Solution:

Image will be added soon

 r = 2 cos θ

Since the equation of the circle in the polar coordinates would be r = 2 cos θ, we make use of the radial stripes having the limits of:


\[\int \int_{R}^{}f(x,y)dxdy\] = \[\int_{-\pi/2}^{\pi/2}\int_{0}^{}\int_{0}^{2 cos\theta}\frac{1}{r}rdrd\theta =\int_{-\pi/2}^{\pi/2}\int_{0}^{}\int_{0}^{2 cos\theta}drd\theta\]


(inner) r from 0 to 2cos θ; (outer) θ from −π/2 to π/2.


Thus, you get:


\[\int \int_{R}^{}f(x,y)dxdy\] = \[\int_{-\pi/2}^{\pi/2}\int_{0}^{}\;1\]


Where the inner integral is 2 cos θ.

FAQs on Double Integral

1. What is a double integral and what does it represent geometrically?

A double integral is a type of definite integral extended to functions of two variables, like f(x, y). Geometrically, it calculates the signed volume of the solid that lies under the surface defined by z = f(x, y) and above a specific region R in the xy-plane. If f(x,y) = 1, the double integral simply gives the area of the region R.

2. How does a double integral differ from a single integral?

The primary difference lies in the dimensions they operate on. A single integral, ∫f(x)dx, is used to find the area under a curve in a 2D plane by integrating over a 1D interval. A double integral, ∬f(x,y)dA, is used to find the volume under a surface in 3D space by integrating over a 2D region. Essentially, we move from finding area to finding volume.

3. What are the main applications of double integrals in real life?

Double integrals are crucial for solving problems where a quantity is distributed over an area. Key applications include:

  • Calculating Volume: Finding the volume of objects with irregular shapes, like the amount of earth in a hill.
  • Finding Mass: Determining the total mass of a flat plate (lamina) with non-uniform density.
  • Centre of Mass: Locating the balancing point of a two-dimensional object.
  • Average Value: Calculating the average value of a function over a region, such as the average temperature across a metal plate.

4. What is the general method for evaluating a double integral in Cartesian coordinates?

A double integral is evaluated as an iterated integral. The process involves:
1. Defining the Region: Clearly sketch the region of integration in the xy-plane and determine its boundaries.
2. Setting Limits: Choose an order of integration (either dx dy or dy dx). The limits of the inner integral can be functions, while the limits of the outer integral must be constants.
3. Integrating: Evaluate the inner integral first, treating the other variable as a constant. Then, integrate the resulting expression with respect to the outer variable.

5. What are some key properties of double integrals?

Double integrals share several properties with single integrals, which simplify calculations:

  • Linearity: The integral of a sum of functions is the sum of their integrals. Constants can be factored out. ∬[c1*f(x,y) + c2*g(x,y)]dA = c1*∬f(x,y)dA + c2*∬g(x,y)dA.
  • Additivity: If the integration region R can be split into two non-overlapping regions R1 and R2, the integral over R is the sum of the integrals over R1 and R2.

6. Why is a double integral used to calculate volume and not just area?

While a double integral can find area (when the function is f(x,y)=1), its main power is in calculating volume. Think of the term dA (or dx dy) as an infinitesimally small patch of area on the floor (the xy-plane). The function value, f(x, y), represents the height of the surface above that tiny patch. The product f(x, y)dA gives the volume of a very thin column. The double integral sums up the volumes of all these infinite, thin columns across the entire region to give the total volume.

7. Does the order of integration (dx dy versus dy dx) affect the result of a double integral?

According to Fubini's Theorem, if the function is continuous over a closed, bounded region, the order of integration does not change the final answer. The result will be the same whether you integrate with respect to x first and then y, or vice versa. However, the choice of order can be strategic. One order might result in a much simpler integral to solve than the other, so choosing the more convenient order is a key problem-solving skill.

8. Can the value of a double integral be negative, and what would it mean?

Yes, the result of a double integral can be negative. The double integral calculates the signed volume. If the surface z = f(x, y) is above the xy-plane (i.e., f(x,y) > 0), the volume is positive. If the surface is below the xy-plane (f(x,y) < 0), the volume is negative. If the surface is partly above and partly below, the double integral calculates the net volume: the volume of the part above the plane minus the volume of the part below it.

9. At what level of mathematics are double integrals typically studied?

Double integrals are a core topic within Multivariable Calculus (often called Calculus III). In the context of Indian education, this subject is generally introduced in the first or second year of undergraduate engineering programs (like B.E. or B.Tech.). While not part of the standard CBSE Class 12 syllabus, the foundational concepts of integration from Class 12 are essential for understanding it. Advanced concepts related to double integrals may also appear in higher-level mathematics for competitive exams like JEE Advanced.