Associated Topics || Dr. Math Home || Search Dr. Math

### Volume Using Cross Sections

```
Date: 05/28/2001 at 00:56:34
From: Purvi
Subject: Volume using cross-sections

I need to find the volume of the region between  y = |x| and
y = -|x|+6 with cross sections that are equilateral triangles and
perpendicular to the x-axis.

The integral I came up with is S from -3 to 3 of (-2|x|+6)*(.5|x|)dx
but I'm not so sure about it. I might have to change my graph a little
because the volume of the region must be no less than 64 cubic inches
and no more than 512 cubic inches. I'm trying to make a pyramid out of
this because with this project we get graded on creativity and
complexity. I would really appreciate it if you are able to help me.
```

```
Date: 05/28/2001 at 17:45:32
From: Doctor Jaffee
Subject: Re: Volume using cross-sections

Hi Purvi,

First, let's talk geometry; specifically equilateral triangles.
Suppose you have an equilateral triangle whose sides each have length
's'.  If you draw the altitude from one of the vertices, you split the
triangle into two 30-60-90 triangles. You recall that the shorter leg
is half as long as the hypotenuse and the longer leg is sqrt(3) times
as long as the shorter leg. So, using this information you should be
able to write a formula for the area in terms of 's'.

Now, if you look at the region bounded by y = 6 - |x| and y = |x|,
you can see that it is symmetrical to the y-axis. In the first
quadrant your two functions can be expressed simply as y = 6 - x and
y = x, and as you figured out, they intersect at (3,3).

So, my suggestion is that you let 0 and 3 be the limits of
area of the equilateral triangle where s = 6 - 2x.

Give it a try and if you want to check your solution with me, write
back. If you are having difficulties, let me know and show me what you
have done so far, and I'll try to help you some more.

Good luck.

- Doctor Jaffee, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
College Calculus
High School Calculus

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search