In many of the following problems, it will be essential to have a calculator or comp...

In many of the following problems, it will be essential to have a calculator or computer available. You may use a software package† or write a program for solving initial value problems using the improved Euler’s method algorithms on pages 127 and 128. (Remember, all trigonometric calculations are done in radians.)

Since the integral y (x) with variable upper limit satisfies (for continuous f ) the initial value problem

any numerical scheme that is used to approximate the solution at x = 1 will give an approximation to the definite integral

Derive a formula for this approximation of the integral using

(a) Euler’s method.

(b) the trapezoid scheme.

(c) the improved Euler’s method.