Question

How do you find the Riemann sum for `f(x) = 4 sin x`, `0 ≤ x ≤ 3pi/2`, with six terms, taking the sample points to be right endpoints?

Answer 1

`(pisqrt2)/2`

Explanation:

`n=6, a=0, b=(3pi)/2`

`Deltax=(b-a)/n=((3pi)/2)/6=(3pi)/12=pi/4`

`S=sum_(i=1)^6 f(x_i)Deltax`

`S=Deltax(f(x_1)+f(x_2)+f(x_3)+f(x_4)+f(x_5)+f(x_6))`

`x_i=a+i*Deltax=0+i*Deltax=i*Deltax`

`f(x_1)=f(pi/4)=4sin(pi/4)=4 sqrt2/2=2sqrt2`
`f(x_2)=f(2pi/4)=4sin(pi/2)=4`
`f(x_3)=f(3pi/4)=4sin(3pi/4)=4 sqrt2/2=2sqrt2`
`f(x_4)=f(4pi/4)=4sin(pi)=0`
`f(x_5)=f(5pi/4)=4sin(5pi/4)=4 (-sqrt2/2)=-2sqrt2`
`f(x_6)=f(6pi/4)=4sin(3pi/2)=4 (-1)=-4`

`S=pi/4(2sqrt2+4+2sqrt2+0-2sqrt2-4)=pi/4 2sqrt2=(pisqrt2)/2`