[Double integral] Area of a triangle

In summary, the problem involves calculating the double integral of (x-y)*|ln(x+2y)| over the triangle with corners at (0,0), (1,1), and (-3,3). The suggested approach is to use substitution to simplify the integral, with u=x-y and v=x+2y. This results in a rectangular triangle with coordinates (0,0), (2,0), and (0,2) in the new coordinate system. The integral can then be set up again using these new coordinates.
  • #1
6
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Hi! I'm stuck with the following problem:

-----------------------------------
Calculate

∫∫ (x-y)*|ln(x+2y)| dxdy

where D is the triangle with corners in the coordinates (0,0), (1,1) and (-3,3)
-----------------------------------

I get the following lines that forms the triangle: y=-x, y=x and y=-1/2*x+3/2

BX4tVDi.jpg


Im thinking that I start with substitution: (x-y)=u and x+2y=v

Which gives me ∫∫ u*|ln(v)| dudv

After that I don't know how to continue. I would like to know which new "intervals" I should do the integral between when I am in my new u- and v-domain?
 
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  • #2
Hatmpatn said:
Im thinking that I start with substitution: (x-y)=u and x+2y=v
That gives ugly borders.
I would use u=x-y and v=x+y then your triangle stays a nice rectangular triangle.

In either case, find the coordinates of the corners in the new coordinate system and then set up the integral again.
 

What is a double integral?

A double integral is a type of mathematical operation that calculates the area under a two-dimensional function. It is represented by two integral signs and involves evaluating the function with respect to two variables.

How is a double integral used to find the area of a triangle?

A double integral can be used to find the area of a triangle by integrating a function over the region of the triangle. This is typically done by dividing the triangle into smaller, simpler shapes and then integrating over each of these shapes.

What is the formula for finding the area of a triangle using a double integral?

The formula for finding the area of a triangle using a double integral is:

A = ∫∫ f(x,y) dx dy

where f(x,y) represents the function that defines the triangle and the integral is evaluated over the region of the triangle.

Can a double integral be used to find the area of any type of triangle?

Yes, a double integral can be used to find the area of any type of triangle, including equilateral, isosceles, and scalene triangles. It can also be used to find the area of triangles with curved sides or irregular shapes.

Are there any limitations to using a double integral to find the area of a triangle?

One limitation of using a double integral to find the area of a triangle is that the function defining the triangle must be known. Additionally, the integration process can be complex for more complicated shapes, making it difficult to find an exact solution. In these cases, numerical methods may be used instead.

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