Shortcuts to find a solution to a double integral

In summary, the conversation discusses the value of an integral equaling 0 and the possibility of using symmetries or odd properties to spot the answer. One person points out that the given boundaries are incorrect and suggests splitting the region of integration into two parts. The other person acknowledges the mistake and explains their approach for calculating the integral.
  • #1
Amaelle
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Homework Statement
Let T be the triangle of vertices (0,0),(2,2),(-4,0) . compute the integral ∫∫xydxdy over the region T
Relevant Equations
0<=y<=(x+4)/3
-4<=x<=2
I know the value of this integral is equal to 0, but I would like to see if there is any tricks to spot this answer using symmetries or even odd propreties?
Thanks in advance
 
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  • #2
This isn't a trick but I thought I'd just point out that the boundaries that you gave aren't right (i.e. yours is for a right angled triangle); you need to split the region of integration into ##0 \leq y \leq \frac{x+4}{3}##, ##-4 \leq x \leq 0## as well as ##x \leq y \leq \frac{x+4}{3}##, ##0 \leq x \leq 2##.
 
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  • #3
yes thanks a lot , you are right the point is that I used a big a triangle with the following boundaries ( and calculated the integral over it) and then substructed the inetgral over a s smaller integral with the following boundaries
0<x<2 and 0<y<x
 
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1. How do I determine the limits of integration for a double integral?

The limits of integration for a double integral depend on the region of integration and the order of integration. To determine the limits, you can draw a diagram of the region and identify the intersection points of the curves or surfaces that define the region. You can also use the given equations to solve for the limits.

2. What is the difference between a double integral and a single integral?

A single integral calculates the area under a curve in one dimension, while a double integral calculates the volume under a surface in two dimensions. In other words, a single integral is used to find the area of a curve, while a double integral is used to find the volume of a solid.

3. How do I choose the order of integration for a double integral?

The order of integration can be chosen based on the given equations or the shape of the region. Generally, it is easier to integrate with respect to the variable that has the simpler limits of integration or the simpler integrand. You can also use symmetry to determine the order of integration.

4. What are some common techniques for solving double integrals?

Some common techniques for solving double integrals include changing the order of integration, using polar coordinates, and using the properties of symmetry. You can also use substitution or integration by parts to simplify the integrand.

5. Are there any shortcuts for solving double integrals?

Yes, there are some shortcuts for solving certain types of double integrals. For example, if the integrand is a product of two functions, you can use the Fubini's theorem to split the integral into two single integrals. You can also use Green's theorem or the Divergence theorem to convert a double integral into a line or surface integral, respectively.

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