Expert: Paul Klarreich - 11/29/2007

Question
Hi I was wondering if you could help me out with this problem.  I am having some difficulty with it.

It says: One of the most important questions that arise in mathematics is how to find ares of objects.  One way of approximating the area of a region is by making rectangles.  Take for example the graph y=cosx on the interval from [0,pi/2].  Create four rectangles at exact values to determine the area between the graph of y=cosx and the x-axis on the interval [0,pi/2].  Use the diagram as a guide, and yes the fourth rectangle has area 0.
              
              y-axis
                 |
            (0,1)|___
                 |   |____
                 |   |    |____
                 |   |    |    |
                 |   |    |    |
   --------------------------------.---       x-axis
                 | pi/0 pi/4 pi/3 (pi/2,0)
                 
From the point (0,1) it connects in an arc shape down to (pi/2,0).

-Your help is greatly appreciated.  

Answer
Questioner:   Nick
Category:  Advanced Math
Private:  No
 
Subject:  Trigonometry
Question:  Hi I was wondering if you could help me out with this problem.  I am having some difficulty with it.

It says: One of the most important questions that arise in mathematics is how to find ares of objects.  One way of approximating the area of a region is by making rectangles.  Take for example the graph y=cosx on the interval from [0,pi/2].  Create four rectangles at exact values to determine the area between the graph of y=cosx and the x-axis on the interval [0,pi/2].  Use the diagram as a guide, and yes the fourth rectangle has area 0.
             
             y-axis
                |
           (0,1)|___
                |   |____
                |   |    |____
                |   |    |    |
                |   |    |    |
  --------------------------------.---       x-axis
                | pi/0 pi/4 pi/3 (pi/2,0)
                
From the point (0,1) it connects in an arc shape down to (pi/2,0).

-Your help is greatly appreciated
...............................................

IF this problem means what I think it does, it belongs in a calculus course, not a trig course.   

It makes no sense to say 'Create rectangles ... to DETERMINE the area.'  If you don't know what the area is, how do you decide where to put the rectangles.

But.....

How did you decide those values:  pi/0 <Did you mean to write that?>, pi/4, pi/3?

The usual scheme for APPROXIMATING the area is:

1. Subdivide your interval [0,pi/2] into four pieces.  [You said four, didn't you?]
That would make division points at  pi/8, 2pi/8, 3pi/8, so your pieces are:

 k    Piece[k]
------+-----------------------
 1     [0,pi/8]
 2     [pi/8,2pi/8]
 3     [2pi/8, 3pi/8]
 4     [3pi/8,4pi/8]

Now choose a point in each subinterval:  You must decide how to do this, choosing from these:

UPPER SUM: Choose x to give the largest cos x.  [LEFTMOST]

LOWER SUM: Choose x to give the smallest cos x.  [RIGHTMOST]

and many other ways.  We'll try LOWER SUM here, because.....
Now we finish the table:

 k    Piece[k]           x[k]    cos(x[k])   
------+------------------------------------
 1     [0,pi/8]         pi/8        ??
 2     [pi/8,2pi/8]     2pi/8       ??
 3     [2pi/8, 3pi/8]   3pi/8       ??
 4     [3pi/8,4pi/8]    4pi/8       ??

You will do the rest -- compute cos(x[k]), multiply by the base of each rectangle, which is pi/8, to get AREA[k], then add them up.

Happy calculating.