Double Integration

[Syed Zain Uddin]

I need a little help with finding upper and lower limits of the below equation using four vertices of a Trapezoid. The four vertices are upper left(1,3), upper right(5,3), lower left(2,1), and lower right(4,1).
The equation is: [FONT=MathJax_Size1]∬ xy [FONT=MathJax_Math-italic]d[/FONT][FONT=MathJax_Math-italic]A[/FONT]
[/FONT]
I know that [FONT=MathJax_Math-italic]d[FONT=MathJax_Math-italic]A[/FONT] = [/FONT][FONT=MathJax_Math-italic]dx.dy[/FONT] and I can double integrate the equation on my own after I've found the limits, finding limits is where I'm having problem at, so if anyone can help me calculate the limits or point me in the right directions to do that I'd really appreciate it. Thank you.

 

[Deleted member 4993]

I need a little help with finding upper and lower limits of the below equation using four vertices of a Trapezoid. The four vertices are upper left(1,3), upper right(5,3), lower left(2,1), and lower right(4,1).
The equation is: [FONT=MathJax_Size1]∬ xy [FONT=MathJax_Math-italic]d[/FONT][FONT=MathJax_Math-italic]A[/FONT]
[/FONT]
I know that [FONT=MathJax_Math-italic]d[FONT=MathJax_Math-italic]A[/FONT] = [/FONT][FONT=MathJax_Math-italic]dx.dy[/FONT] and I can double integrate the equation on my own after I've found the limits, finding limits is where I'm having problem at, so if anyone can help me calculate the limits or point me in the right directions to do that I'd really appreciate it. Thank you.

I'll break-up the region into three parts:

One rectangle - (2,1), (4,1), (4,3) & (2,3) - limits for this is trivial.

One triangle is bounded by (2,1),(2,3) and (1,3)

For this you would need to calculate the equation of the line joining (1,3) and (2,1)

Similarly, other triangle is bounded by (4,1),(5,3) and (4,3)

For this you would need to calculate the equation of the line joining (4,1) and (5,3).

If you need more assistance, please reply with listing of your work.

 

[Syed Zain Uddin]

Will you please just tell the limits of x and y ?

 

[Syed Zain Uddin]

As you said that..
I have done..
equation of the line joining (1,3) and (2,1) is y=-2x+5
and
equation of the line joining (4,1) and (5,3) is y=2x-7

So what further I do? to find out the limts of x and y?

 

[Deleted member 4993]

As you said that..
I have done..
equation of the line joining (1,3) and (2,1) is y=-2x+5
and
equation of the line joining (4,1) and (5,3) is y=2x-7

So what further I do? to find out the limts of x and y?

Now you think....

The first region (on the left-side) is bounded by [y=-2x+5 and y = 3] and [x =1 and x=2]

What limits does that generate?

What are the limits for the rectangular zone?

What are the limits for the other triangular zone?

 

[Syed Zain Uddin]

Double integration

So, the first region (on the left-side) limits = 1 ≤ y ≤ 2 and 3 ≤ y ≤ -2x+5

Am i right or wrong ?

 

[Syed Zain Uddin]

Now you think....

The first region (on the left-side) is bounded by [y=-2x+5 and y = 3] and [x =1 and x=2]

What limits does that generate?

What are the limits for the rectangular zone?

What are the limits for the other triangular zone?

Oh sorry correction...

So, the first region (on the left-side) limits = 1 ≤ x ≤ 2 and 3 ≤ y ≤ -2x+5

and the limits for the rectangular zone = 2 ≤ x ≤ 4 and 1 ≤ y ≤ 3

Am i right or wrong ?

 

[Deleted member 4993]

Oh sorry correction...

So, the first region (on the left-side) limits = 1 ≤ x ≤ 2 and 3 ≤ y ≤ -2x+5

and the limits for the rectangular zone = 2 ≤ x ≤ 4 and 1 ≤ y ≤ 3

Am i right or wrong ?

Almost:

So, the first region (on the left-side) limits = 1 ≤ x ≤ 2 and -2x+5 ≤ y ≤ 3

and the limits for the rectangular zone = 2 ≤ x ≤ 4 and 1 ≤ y ≤ 3

 

[Deleted member 4993]

And now set up he limits for the third region (triangle on the right)