# Applications of Method of Exhaustion

1. Jan 29, 2013

### number0

1. The problem statement, all variables and given/known data

Show that the area of two triangles with the same base and height can be
approximated arbitrarily closely by the same set of rectangles, differently
stacked.

See attachment for picture!

2. Relevant equations

Use method of exhaustion (NOT proof by exhaustion).

3. The attempt at a solution

The book I have does not clearly explain what the method of exhaustion is. I looked online I still cannot find out how it is used except for the famous case where Archimedes used it to estimate the area of the circle or something related to that manner.

What I did was I ASSUMED that you can use a set of rectangles, call it M, to approximate the area of Triangle A.

Let N be define as the set of rectangles from M, but differently stacked.

Let Triangle B be define as a Triangle that has the same base and height as Triangle A.

Want to show that: Area(N) approximate Area(Triangle B).

By assumption, Area(M) approximates Area(Triangle A)

But, Area(Triangle A) = Area (Triangle B)

=> Area(M) approximates Area(Triangle B)

But, Area(M) = Area(N)

=> Area(N) approximates Area(Triangle B)

End.

Since this problem is listed under the Method of Exhaustion section, I am certain that I did not use Method of Exhaustion and thus I am pretty sure I did something wrong or that I misinterpret the question or made a bold assumption or etc.

Anyone have an advice on how to implement the Method of Exhaustion for this particular problem or for any other problem in general? Thanks.

* Side note: From what I gathered so far, Method of Exhaustion assumes double contradiction to prove the equality. For instance, show that A < C is false and that C > A is false so that A = C is true. It has been noted that Method of Exhaustion is "like" taking the limit as n->infinity and is considered to be the precursor to calculus. I also have another homework problem that is asking for Method of Exhaustion but it is not related to this one.