Multiple Integrals Tutorial and Double Integral Solutions with Examples

Multiple Integrals

Multiple Integrals means during integration of one variable can be extended to the functions of more than one variable. The generalizations of define integrals is known as Multiple integral. Multiple integral covers several of topics such as:

1. Double Integrals
2. Iterated Integrals
3. Double Integrals in polar coordinates
4. Triple Integrals
5. Triple integrals in cylindrical coordinates

Note: Triple Integrals in spherical coordinates and Double Integrals over general regions.

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Best Quality Double Integral Problems and Detailed Solutions

Notation of Double Integral





This shows the double integral, over the reason of z of f(x,y).

Working Method of double integral








First Integral (A), Integrate with respect to y and x is assumed as a constant and evaluate the result between the limit y = f1(x) and y f2(x). After this we integrate the result of (A) with respect to x between the limit x=a and x=b.

Examples of Double Integrals
(1)








Step - 1
First we Integrate with respect to y and x is assumed as a constant between the limit y=0 and y=x.



























Example-2

y = 3x

y = 5-2x

y = 0


























Solving these three equations

y = 3x

y = 5 – 2x

y = 0

we get lines cut at (0,0) , (5/2, 0) and (1, 3)

thus limit of x will vary from x = 0 to 5/2.

And limit of y will vary from y = 3x to y = 5 – 2x














Area A = 25/8 Ans.

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