# Thread: really tricky multiple integral word problem

1. ## really tricky multiple integral word problem

So I've been trying to figure out this question for the past couple of days and have had no luck with it.

"A region in space when viewed from 3 different views, looks like a circle, looks like a square and looks like a triangle. Describe this object in words or pictures, and then use multiple integration to determine its volume"

Any two shapes would be easy enough to figure out, square + triangle is a pyramid, circle + triangle is a cone and so on, but adding that 3rd shape really throws me off.

2. Take your basic can of tomato soup.
Use your best can opener and remove BOTH ends.

Look through the tube - circles.

Hold it sideways, horizontal rotation of 90º. It's probably a rectangle. A little snipping will get squares.

Look down from the top. Still a square. Squish one end. A triangle! Again, since squishing deforms the can, a little snipping may be required to get it square again.

3. Hello, numberonenacho!

A region in space when viewed from 3 different views,
looks like a circle, looks like a square and looks like a triangle.
Describe this object in words or pictures,
and then use multiple integration to determine its volume.

Consider a right circular cylinder with diameter $a$ and height $a$.

Stand it on one of its circular bases.

Looking down, we see a circle.
Looking from the side, we see a square . . . like this:
Code:
*---------------*
|               |
|               |
|               |
a |               |
|               |
|               |
|               |
*---------------*
a

Make two diagonal cuts as shown below.
Code:
* - - - * - - - *
:      / \      :
:     /   \     :
:    /     \    :
:   /       \   :
:  /         \  :
: /           \ :
:/             \:
*---------------*

Looking down, we see a circle.
Looking from the right (or left), we see a square.
Looking from the front, we see a triangle.