How do you use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpson’s Rule to approximate the given integral #(1+cos(x))^(1/3)# from #[pi/2,0]?
Trapezoidal : 0.4597
I am reading this as
Pick 5 points (4 strips) x=0, π/8, 2π/8, 3π/8. 4π/8. Each strip is of width h=π/8.
Evaluate f(x) at each point:
and calculate function at x=π/16, 3π/16, 5π,16, 7π/16:
r 1 .1587
For trapezium rule: I=h(a+2b+2c+2d+e)/8 =0.4597
For Simpson's rule: I=h(a+6b+4c+6d+e)/18 =0.4622
For mid-point rule: I=h(p+q+r+s)/4=0.4613
Notice how all the answers are just the weighted averages of the various y-values (e.g. 1+6+4+6+1=18 etc), the different rules giving different weights. One could add also an even cruder "left value' and 'right' value estimate which, because the function is monotonic, would give upper and lower bounds.