Chapter 8.8, Problem 74E

Sine integral The theory of diffraction produces the sine integral function $\text{Si}(x)={\displaystyle {\int }_0^x\frac{\sin t}{t}}dt$. Use the Midpoint Rule to approximate Si(1) and Si(10). (Recall that $\underset{x\to 0}{\lim }\frac{\sin x}{x}=1$.) Experiment with the number of subintervals until you obtain approximations that have an error less than 10⁻³. A rule of thumb is that if two successive approximations differ by less than 10⁻³, then the error is usually less than 10⁻³.