Use the trapezoidal rule, the midpoint rule, and simpson's rule to approximate the given integral with the specified value of n. (round your answers to six decimal places.) π/2 0 3 1 + cos(x) dx, n = 4

Split up the integration interval into 4 subintervals:

The left and right endpoints of the -th subinterval, respectively, are

for , and the respective midpoints are

We approximate the (signed) area under the curve over each subinterval by

so that

We approximate the area for each subinterval by

so that

We first interpolate the integrand over each subinterval by a quadratic polynomial , where

so that

It so happens that the integral of reduces nicely to the form you're probably more familiar with,

Then the integral is approximately

Compare these to the actual value of the integral, 3. I've included plots of the approximations below.

b) well because you get well