Solution to Exercise 12.3-1

© 2005 by Karl Hahn 		KCT logo (click for Home Page)
The problem is to find the area between the curve,  y = x4 - 3x3 + 3x2 - 2x + 2  and the line,  y = x.  Again the first thing to do is take the difference of the two functions. 
  x4 - 3x3 + 3x2 - 2x + 2
                    x
                         
  x4 - 3x3 + 3x2 - 3x + 2
As I said in the main text, the resulting difference is easy to factor. Look at how it's done before you look at the graph. If there are easy roots to this, they must divide the constant coefficient, which is 2. So the candidates for the roots are ±1 and ±2. You can either try the candidates in the polynomial to see if you get zero, or you can try doing polynomial long division to see if you get a remainder. Using the first method you can see that -1 and -2 can't work because they both give positive values for each term in the polynomial -- hence the sum of the terms can't possibly be zero. When you put 1 in for x, that's the same as simply adding up the coefficients, which does indeed give zero. So  x = 1  is a root. And with just a little pencil-pushing you can see that  x = 2  is also a root. Are there any other roots? Since this is a fourth degree polynomial, we know it can have as many as four roots. But now that we know two of them, we can divide them out and be left with a quadratic: 

          x3 - 2x2 +  x  - 2
         _________________________
   x - 1 )x4 - 3x3 + 3x2 - 3x  + 2
          x4 -  x3
                  
              -2x3 + 3x2
              -2x3 + 2x2
                        
                      x2 - 3x
                      x2 -  x
                             
                           -2x + 2
                           -2x + 2
                                  
                                 0

          

         x2 +  0  +  1
        ___________________
  x - 2 )x3 - 2x2 +  x  - 2
         x3 - 2x2
                 
                0 +  x
                      
                     x  - 2
                     x  - 2
                           
                          0

So when you divide the two known roots out, you get  x2 + 1,  which has no real roots. We are now finished extracting roots -- on to the integration.

Using the method described in the main text, we simply integrate the difference function from one root to the other. 

  A  =

  2
  
  1

 x4 - 3x3 + 3x2 - 3x + 2 dx


  =

  

 x5
    -
  5

 3x4
     +
  4

 x3 -

 3x2
     + 2x
  2

 2
 
 1

  =

  2
    -
  5

 19
     =  -
 20

 11
   
 20

As with the last exercise, we got the difference backward again, and that has led to a negative area (note in the graph that the line is consistently greater than the curve in the region where area is enclosed, but we took the curve minus the line). So the true result is  A = 11/20. 

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