Solutions Manual to accompany An Introduction to Numerical Methods and Analysis, 3rd Edition

CHAPTER 5NUMERICAL INTEGRATION

5.1 A REVIEW OF THE DEFINITE INTEGRAL

Exercises:

1. Basic properties of the definite integral show that it is a linear operator, i.e., it distributes across sums and multiplication by constants:
   

   Prove that if

   

   then is also linear:

   

   Solution: This is a straight‐forward computation:

   
2. Assume that the quadrature rule integrates all polynomials of degree less than or equal to exactly:
   

   for all . Use this to prove that, for any integrand , the error is equal ...