Question

What is the value of
\[\int\limits_{}^{} \int\limits_{}^{} \int\limits_{R}^{} \cos(x + y + z) ~dx~dy~dz\]
\[\ \text{Where R is the region bounded by:}\]
\[\ x \ge 0 , y \ge 0 , z \ge 0 ~~\text{and}~~~x+y+z \le 2 \pi \]

Answer 1

._. i feel so stupid looking at this

Answer 2

can you describe the solid?

Answer 3

That's all the info given.

Answer 4

looks like a triangle prism

Answer 5

from your drawing

Answer 6

oh wait that's a pyramid.

Answer 7

z ranges from 0 to the plane yes?

Answer 8

yeah

Answer 9

what about x and y?

Answer 10

the same?

Answer 11

what is the equation that describe the line?

Answer 12

\[x + y + z \le 2{\pi}\] ?..
I don't fully understand how to do this ... Could you give an approach without using a graph ? xD

Answer 13

Well, most of the time, it's impossible to describe a solid without drawing, but in this case, it's possible (I guess because I already had this image in my mind, so I can do it without drawing)

the plane intersect the xy-plane at the line x + y = 2pi (just plug 0 in for z)

Answer 14

ok

Answer 15

so y = 2pi - x yes?

Answer 16

Yeah.

Answer 17

x ranges from 0 to 2pi
y ranges from 0 to 2pi - x
z ranges from 0 to 2pi - x - y

those are the limits of integration. From here on, it's all computation, which I sense it's gonna be quite tedious. If you already how to integrate, just use wolfram alpha

Answer 18

yeah i know how to integrate from here , thanks for the clarification on the limits

Answer 19

yw

Answer 20

btw, answer turns out to be 2pi
http://www.wolframalpha.com/input/?i=integrate+cos%28x%2By%2Bz%29+dz+dy+dx%2C+x%3D0..2pi%2C+y%3D0..2pi-x%2C+z%3D0..2pi-x-y

Answer 21

I tried putting it in , but I got exceeded time limit.. xD do you have wolframpro?

Answer 22

no, i just use the website.