# Math Help - Double Integrals over General Regions: Triangle

1. ## Double Integrals over General Regions: Triangle

The problem is this:

Evaluate the double integral. , D is the triangular region with vertices (0,0), (2,4), and (6,0).
Solution Attempt:
Line 1: y=2x
Line 2: y=-x+6
0=<x=<6

Integrating ye^x dy dx yields 1/2y^2 e^x and when evaluated from 2x (bottom) to -x+6(top) it yields 1/2 e^x ((-x+6)^2 -4x^2); integrating this with respect to x yields e^x (-14+2x+x^2) and evaluating this from 0 to 6 yields -21-51e^6.

This is a problem given online and says this is incorrect but i've done this by hand and in maple. Please point out what i've done wrong. Thank you.

2. Originally Posted by jdj0202
The problem is this:

Evaluate the double integral. , D is the triangular region with vertices (0,0), (2,4), and (6,0).
Solution Attempt:
Line 1: y=2x
Line 2: y=-x+6
0=<x=<6

Integrating ye^x dy dx yields 1/2y^2 e^x and when evaluated from 2x (bottom) to -x+6(top) it yields 1/2 e^x ((-x+6)^2 -4x^2); integrating this with respect to x yields e^x (-14+2x+x^2) and evaluating this from 0 to 6 yields -21-51e^6.

This is a problem given online and says this is incorrect but i've done this by hand and in maple. Please point out what i've done wrong. Thank you.

When considering the y limits, draw a line that passes straight through the region from negative values of y to postive values. Clearly, that line ENTERS the region through the x axis, but which line does it EXIT the region through? Well there's one of two lines it exists through... it can exit through either $y = 2x$ or $y = x - 6$.

Do the same with the x direction. Draw a line from negative values of x to positive values of x that passes straight through the region. You'll see that it passes through ONE line to enter the region ( $y = 2x$) and passes through ONE line to exit the region ( $y = x - 6$).

That tells you that you should probably allow you x limits to be functions of y, and allow you y limits to be the consants. So:

$\frac{y}{2} \leq x \leq y + 6$

$0 \leq y \leq 4$

$\int_0^4 \int_{\frac{y}{2}}^{y+6} ye^{x} \, dx \, dy$