Question

How do you write the Simpson’s rule and Trapezoid rule approximations to the `intsinx/x` over the inteval [0,1] with `n=6`?

Answer 1

Simpson's sule with n=6 is simply:
`int_0^1f(x)=1/18[f(0)+4f(1/6)+2f(2/6)+4f(3/6)+2f(4/6)+4f(5/6)+f(1)]`

Here f(x) is just `sinx/x` and so one must calculate this for x=0, 1/6, ... 5/6, 1 and put the results into the formula.

The Trapezoidal rule is similar:
`int_0^1f(x)=1/12[f(0)+2f(1/6)+2f(2/6)+2f(3/6)+2f(4/6)+2f(5/6)+f(1)]`

Note that, although `f(0)` does not exist, we know that `lim_(xrarr0)f(x)=lim_(xrarr0)sinx/x=1`.

Use `1` where the formula calls for `f(0)`.