Question

Let f(x) = sin(x) on the interval I = [0,π]. Also, let n = 10.

a.) Setting this problem up for a midpoint Riemann sum, determine ∆x and a formula for

x∗k for the interval I and n given above:

Answer 1

∆x=b-a/n

Here [a,b]=[0,π] and n=10

Thus ∆x=π/10.

Now we define our partitions as follows,

[0,π/10] , [π/10,2π/10] , [2π/10,3π/10] , [3π/10,4π/10] , [4π/10,5π/10] , [5π/10,6π/10], [6π/10,7π/10] , [7π/10,8π/10] , [8π/10,9π/10] , [9π/10,π]

Then find mid point of these partitions,

Xi={π/5,3π/20,π/4,7π/20,9π/20,11π/20,13π/20, 3π/2,17π/20,19π/20}

Now find value of function at these points,

f(π/5)=sinπ/5=0.59

sin3π/20=0.45

sinπ/4=0.71

sin7π/20=0.89

sin9π/20=0.99

sin11π/20=0.99 and so on upto

sin19π/20=0.16.

These are values of f(Xi)

Now reimann sum A is given as

A=∆x.

=(0.59+0.45+0.71+0.80+0.99+0.99+...+0.16)π/10.