Computing a double integral with given vertices

[ramb]

1. Homework Statement [/b]

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

$$\int\int_{B}2x+3y dA$$

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.

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

With the first integral I got

$$\dfrac{-651}{8}$$

for the second integral i got

$$\dfrac{-4739}{12}$$

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, $$\frac{11431}{24}$$ 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

 

[LCKurtz]

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.