1. ## triple integral

Determine the volume of the region that lies behind the plane x+y+z=8 and in front of the region in the yz-plane that is bounded by z=(3/2) *sqrt(y) and z=(3/4)*y.

Please teach me how to solve this question by using triple integral.
If posible, please show me how the graph of the solid look like. thank you very much.

2. We're hanging out in the first octant. That does make it easier. There is no secret. You just have to figure it out a piece at a time. For starters, the definition of the y-limits seems to be the trickiest, so we may wish to do that last and cut it down to just numbers, rather than functions.

Here's my first attempt.

$\displaystyle \int_{0}^{4}\int_{\frac{3}{4}y}^{\frac{3}{2}\sqrt{ y}}\int_{0}^{8-z-y}1\;dx\;dz\;dy$

Really, the y-limits were the hardest to determine, but it was only a quick 2D algebra problem.

3. Hi Tkhunny,

Thank you very much.

How do we know that is the first octant?

Also, could you please show me how did you set up the y and x limits? Thank you very much.

4. Originally Posted by kittycat
How do we know that is the first octant?
That's just the sense of the descriptions. "In front of the y-z plane" cuts out half of space. The intersections of $\displaystyle z = \frac{3}{2}\sqrt{y}$ and $\displaystyle z = \frac{3}{4}y$ are in the first quadrant of the y-z plane. (Well, okay, the Origin and the First Quadrant) That's about it. The evidence is sufficient.
Could you please show me how did you set up the y and x limits?
I chose to start with the x-limits because y and z seemed to have more going on. It was just a personal choice. Plus, it was obvious how to express the x-limits after solving the equation of the plane for x.

After that, it is just a 2-dimensional problem in y and z. You should have been able to do that piece last semester.

Personally, I was not particularly confident in my result, so I chopped up the entire first octant and determined all the volume on the Origin side of the given plane. The volume of the whole thing is pretty easy, so when the four more complicated pieces added up to the same, unique answer, I had some sense that I was on the right track.

5. Hi Tkhunny,

" I had some sense that I was on the right track."

Yes, you are. The integral that you set up are correct. Thank you very much for your explanation.

Do you use a graphing tool to graph this 3 dimensional graph? Is it important to master the skill of graphing 3D objects manually?
I am asking you these questions because I am very bad in graphing 3D objects manually. I use graphing tool to graph them. Somehow I worry that I may fail this course because I don't put effort to master the skill of graphing 3D objects without the help of a computer program.

6. Hmpf. I am a very, very bad artist.

I have a nice program that draws 3D images quite readily, but (after a little effort) I could not figure out how to get it to draw more than one equation at a time. In this case, the art work really didn't help all that much.

Seeing it in your mind, whether you can draw it or not, is most important, particularly since the drawings fall apart rather dramatically after you move up from 3D.

You show us the next one. Come on, you can do it!