Double Integral: Find Area of Triangular Vertices

[haris13]

find the area of the double integral ∫∫x + y (is the triangular vertices (0.0) , (2,2) and (4,0)) how to find the values of x and y.

 

[Mark44]

find the area of the double integral ∫∫x + y (is the triangular vertices (0.0) , (2,2) and (4,0)) how to find the values of x and y.

What have you tried? Before we can give you any help, you must have made an effort to solve the problem.

Also, I'm not sure that you understand the problem. I believe that you are to evaluate this integral
$$\int \int_R x + y~dA$$
where R is the triangular region with vertices (0, 0), (2, 2), and (4, 0). If so, the integral probably doesn't represent an area, and you are not supposed to solve for x and y.

 

[Syrus]

You must first integrate with respect to one variable and then to another. First write the equations of the lines of your triangle (i will do this in terms of x). The range of your x values is 0 to 4. Now, y values range from 0 to y=x from x=0 to x=2, and y values range from 0 to y= 4 - x from x = 2 to x - 4.

The integral over the intire area can be separated into two simple double integrals:

the integral from 0 to 2 of the integral from 0 to X of (x+y), with respect to y, with respect to x

PLUS

the integral from 2 to 4 of the integral from 0 to (4-x) of (x +y), with respect to y, with respect to x


I'm sorry, still working on getting the symbols and correct boundaries to look nice, ill edit as soon as possible


And yes, you should have made an effort to solve... This wasn't meant to take anything away from the poster above... forgive me, I am new!

 

[haris13]

thanks for your help. I am fairly new at integrals so i was having a hard time. can you please give me the x and y values of both the intergrals of both parts. i still don't know how we arrive at those values. that's my only concern.

 

[Harrisonized]

There are two ways to do this:

1. make your x-bound a numerical interval [a,b], and make the y-bound lines [y₁=m₁x+b₁, y₂=m₂x+b₂]
2. make your y-bound a numerical interval [a,b] and make the x-bound lines, [x₁=m₁y+b₁, x₂=m₂y+b₂]

You should try both ways just to see that they result in the same value.

 

[Mark44]

thanks for your help. I am fairly new at integrals so i was having a hard time. can you please give me the x and y values of both the intergrals of both parts. i still don't know how we arrive at those values. that's my only concern.

What have you tried? Did you read my post #2?

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