Question

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]?

Answer 1

Trapezoidal : 0.4597
Midpoint: 0.4613
Simpons: 0.4622

Explanation:

I am reading this as `int_0^(pi/2)(1+cos(x))^(1/3)dx`
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:
a 1.2599
b 1.243295648
c 1.1951
d 1.1141
e 1.0000

and calculate function at x=π/16, 3π/16, 5π,16, 7π/16:

p 1.2559
q 1.2235
r 1 .1587
s 1.0612

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.