Question

i seem to have made a mistake somewhere, but i don't see it.....help??

Evaluate the triple integral ∭ExydV where E is the solid tetrahedon with vertices (0,0,0),(9,0,0),(0,6,0),(0,0,7).

Answer 1

here's my work....

Answer 2

That's not quite right. You need to integrate by dragging the entire function across the axis, so think of a sheet sort of moving. Like this: So as you integrate in the y-direction the area under the curve changes, so your limits of integration are really a function of multiple variables now.

Answer 3

Now if you just plain do the integral without the xy part you'll get the volume. But if you integrate with it, you'll be evaluating xy at every point inside giving you some value there. I like to think of it as adding up the temperature at each spot to find the total heat content of the object or something like that. Or perhaps it's the density of the object you're integrating to find the mass of the object if that helps you visualize it.

Answer 4

oh. since i took the integral of a rectangle, can i just take half of what i found since it's a triangle basically?

Answer 5

you could do that if the integrand is constant

Answer 6

grr. taking 5103/2 didn't work either. so i guess i can't just divide by two for a triangle.

Answer 7

not possible to simply take half of rectangle when your integrand is "xy"

you need to setup the bounds for triangle and work it again

Answer 8

try this for dzdydx :

x : 0->9
y : 0->6(1-x/9)
z : 0->7(1-x/9-y/6)

Answer 9

gives a negative integral. i entered the negative and abs value and neighter worked.

Answer 10

im getting 170.1
http://www.wolframalpha.com/input/?i=%5Cint%5Climits_0%5E9%5Cint%5Climits_0%5E%7B6%281-x%2F9%29%7D%5Cint%5Climits_0%5E%7B7%281-x%2F9-y%2F6%29%7D+%28xy%29+dzdydx

Answer 11

ok. that works. how did you pick those bounds? i have to drive home, but i'll be back in like 20 minutes.

Answer 12

you're good with setting up bounds for double integral right ?

Answer 13

yes, i understand how to do those a lot better. still hard, but i can get it better