Computing a double integral with given vertices

In summary, the conversation discusses using a transformation to compute the integral of 2x+3y over a triangular region with specific vertices. The person asking the question attempted to solve it by splitting the triangle into two and using double integrals, but the other person states that this is not the method they were asked to use and suggests looking up the transformation given in the textbook.
  • #1
ramb
13
0
1. Homework Statement [/b]

Use the transformation that takes the unit square to a triangle to compute the integral

[tex]\int\int_{B}2x+3y dA[/tex]

Where B is a triangular region with vertices (0,0), (5,2), and (3,4).


The Attempt at a Solution



What I did was I drew the region on an xy plane, I split the triangle up into two triangles and found my limits of integration by drawing the lines made by connecting the vertices. Because I split the triangle up into two, I needed to add two separate double integrals.

This is what I got.

[tex]\int_{x=0}^{x=3}\int_{y=5x/2}^{y=4x/2}(2x+3y) dydx + \int_{x=3}^{x=5}\int_{y=5x/2}^{y=-x+7}(2x+3y) dydx[/tex]

With the first integral I got

[tex]\dfrac{-651}{8}[/tex]

for the second integral i got

[tex]\dfrac{-4739}{12}[/tex]

I figured that if I add both of them together I would get the volume underneath that region (the whole thing), but the number I got, [tex]\frac{11431}{24}[/tex] seems to large.

I also don't think I'm doing it the method wanted.
Can anyone please direct me where to go from here, I'm somewhat lost.

Thank you
 
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  • #2
I didn't check your work because, as you suspect, that isn't the method you have been asked to use. Your question refers to "the" transformation that takes the unit square to a triangle. I'm guessing your text has given you that example. Look it up and share the equations of that transformation with us and then we can talk.:smile:
 

Related to Computing a double integral with given vertices

What is a double integral?

A double integral is a mathematical concept used in calculus to find the volume under a surface defined by a function. It involves finding the area of a shape in two dimensions by integrating over a region in two variables.

What are the vertices in a double integral?

The vertices in a double integral are the four points that define the boundaries of the region over which the integral is being calculated. These points are typically given as coordinates on a graph.

How do you set up a double integral with given vertices?

To set up a double integral with given vertices, you first need to determine the limits of integration for each variable. This is done by identifying the minimum and maximum values for each variable based on the given vertices. Once the limits are determined, the integral can be set up using the appropriate formula.

What is the importance of computing a double integral with given vertices?

Computing a double integral with given vertices allows for the calculation of the volume under a surface in two dimensions. This is a crucial concept in many fields of science and engineering, as it can be used to solve a wide range of real-world problems.

What are some common methods for computing a double integral with given vertices?

There are several methods for computing a double integral with given vertices, including the rectangular, polar, and parametric methods. The method used will depend on the given problem and the shape of the region being integrated over.

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