What is the answer and how do you solve for it?

Question

If f(x) = 4x² − 3x, 0 ≤ x ≤ 3, evaluate the Riemann sum with n = 6, taking the sample points to be right endpoints.

Answer 1

The formula for a Right Reimann Sum is ∑ⁿ_i=1 f(x_i)Δx where Δx=(b-a)/n=(3-0)/6=(1/2). This means that each interval will have a width of 1/2 with n=6 subintervals from 0 to 3 using the rightmost points in each subinterval:

∫₀³ 4x²-3x dx ≈ ∑⁶_i=1 f(x_i)(1/2)

= (1/2)[f(1/2)+f(1)+f(3/2)+f(2)+f(5/2)+f(3)]

= (1/2)[-0.5+1+4.5+10+17.5+27]

= (1/2)(59.5)

= 29.75

Note that the actual integral can be solved easily:

∫₀³ 4x²-3x dx

= (4/3)x³-(3/2)x² [0,3]

= (4/3)(3)³ - (3/2)(3)²

= (4/3)(27) - (3/2)(9)

= 108/3 - 27/2

= 36 - 27/2

= 72/2 - 27/2

= 45/2

= 22.5

So we can see that 6 subintervals with lengths of 1/2 for a right Reimann Sum is an overestimation because the graph increases on that interval.

Hope this helped!