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### Triple Integrals

```
Date: 12/07/97 at 14:31:28
From: Alicia Sussman
Subject: Triple integrals

Hi again. Thank you for answering my last question, it was extremely

Evaluate /  /  /
|  |  |
|  |  |  y dx dy dz, where T is the solid in the first octant
/  /  /
T      bounded by y=1, y=x, z=x+1 and the coordinate planes.

I think that the constraints would be  1<y<x, and x+1<z<y+1 but I
can't figure out for x. Am I on the right track? I know the answer to
the problem is 11/24. Please let me know.

Thanks again!
```

```
Date: 12/08/97 at 06:51:12
From: Doctor Pete
Subject: Re: Triple integrals

Hi,

To determine the limits, try drawing a picture first. Say you are
looking down on the xy-plane, so that the first octant is in the
upper-right corner (just like in the two-dimensional case).  Then the
region of integration looks like a triangle, with vertices (0,0),
(0,1), and (1,1). You can't see the constraint z = x+1 from this
angle, unfortunately. But what we do see from this view is that
0 < x < 1, and x < y < 1.

Now, to understand how I got this from seeing just a triangle, let's
suppose we pick a point, say p, along the x-axis, between x = 0 and
x = 1. Draw a vertical line parallel to the y-axis, which is the line
x = p. Then it will intersect the edges of the triangle at two points,
namely (p,1) and (p,p). Therefore, the y-coordinates of these two
points are your limits on y; hence 0 < x < 1 and x < y < 1.

As for z, we see that it is bounded below by z = 0 (the xy-plane), and
above by another plane, z = x+1. So this limit is simply 0 < z < x+1.
If you picked any point (p,q,0) in the little triangle we talked
about, then the line parallel to the z-axis from that point is going
to intersect the xy-plane at z = 0, and the plane z = x+1 at p+1.

/x=1 /y=1 /z=x+1
|    |    |      y dz dy dx.
/x=0 /y=x /z=0

Notice I have rearranged the integrals, because both the integrals
over y and z are dependent on the value of x. (Does it matter if I put
y first or z first?  Why?)

When you try to determine limits of a triple integral, it is often
very helpful to draw pictures of what the region looks like when you
see it from different angles. Since not many people can visualize 3-D
objects in their heads, it's much easier to look at 2-D projections on
paper. Notice that I also drew lines intersecting the region T to see
why the limits were the way they were.

-Doctor Pete,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
High School Calculus

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