I would really appreciate the help with this For a known fun

I would really appreciate the help with this.

For a known function z(t) (blue line) a few points (red circles) were selected to find a numerical approximation of the integral of the function for the region t=0 to 5. The red circles are given by the points: 1. Approximate Integrate limit between 0 to 5 z(t)dt using a. Left Riemann Sum (by hand) b. Right Riemann Sum (by hand) c. Trapezoidal Method (by hand and using trapz function in Matlab) d. Simpson?s 1/3 and 3/8 Rule (by hand, if applicable!)

Solution

a) interval = 0.0833

left integral = sum(f(x)dx)

= sum(0.8333(0+*0.3018+0.5247+0.5130+0.3964+0.2692))

=1.6708

b) interval = 0.0833

right integral = sum(f(x)dx)

= sum(0.8333*(0.3018+0.5247+0.5130+0.3964+0.2692+0.1684)

=1.8111

c) trapezoidal rule

sum(o.8333/2(0+2*0.3018+2*0.5247+2*0.5130+2*0.3964+2*0.2692+0.1684)

=0.7410

d) simpsons 1/3 rule

= sum(0.8333/3(0+(4*(0.3018+0.5130+0.2692)+2*(0.5247+0.3964)+0.1684))

= 1.7629