Integration Two Variables

Integration is an important concept in mathematics and, together with differentiation, is one of the two main operations in calculus. The term integral may also refer to the notion of antiderivative, a function F whose derivative is the given function ƒ.The integration process with two variables involves the operation of performng integration with different variables for varying the process deal with the article.
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Examples to Explain "integration Two Variables "

Understanding Solving for Variables is always challenging for me but thanks to all math help websites to help me out.

Consider the double integral form eith two variables x and y  `int_0^1` `int_2^1` (y2 + yx3) dx dy

Solution:

`int_0^1` `int_1^2` (y2 + yx3) dx dy  =  `int`  `[y^2x + y x^4/4]_1^2 dy`

we integrate the given function with respect to y in the above step..

=   `int_0^1`  `[y^2x + y x^4/4]_1^2 dy`

=  `int_0^1``[y^2(2) + y (2)^4/4] dy`  

=  `int_0^1` `[y^2 + 15 y /4] dy`    

= `[y^3 + 15 y^2 /8] `    

=  `[ 1/3 + 15/8 ]` + c

In this problem we execute the flow of every step with different variables.
Problems to Explain "integration Two Variables "

Integrate the function given

`int`(x5y+2x5y+3y )dx

Solution:

`int`(x5y+2x5y+3y )dx= `int`x5 ydx + `int`2x5 ydx +`int`3y dx

`int`(x5y+2x5y+3y )dx=y`int`x5 dx + y `int`2x5 dx + y`int`3 dx

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Here one doubt arises to every person that two variables x and y are present  but only dx is present .The solution is that y is treated as a constant here. We get

`int`(x5y+2x5y+3y )dx =  y ( x5/5 )+ y( 2 x5/5 ) +y3x

= (`(3x^5y)/5` )+ 3x y+ c

= `(3x^5y)/5 + 3xy +c`       is the required answer.

Integrate the function given

`int`122yx2 + exydx

Solution:

`int`122yx2 + exydx = `int`122yx2 dx +  `int`exy dx

`int`122yx2 + exydx =y `int`122x2 dx +  `int`exy dx

The same concept in the above problem exist here

We get

`int`122yx2 + exydx =122y (x3 / 3) + exy /y

=`(122yx^3)/3+ e^(xy)/y + c`             is the required answer.

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