How to compute left, right, & midpoint Riemann sums when n value is large (n=25, n=100)

Question

f(x)= x²+1 on the interval [1,4]

Compute the Riemann sums for L₂₅, R₂₅, M₂₅, L₁₀₀, R₁₀₀, M₁₀₀

_{What is the exact area of the curve?}

I managed to get L₅, R₅, & M₅, but am very confused on how to calculate the Riemann sums when n=25 or n=100

Please help, I just want to be able to understand how to do this step-by-step.

Answer 1

Find Δx = (b-a)/n = (4-1)/n

Expand the Riemann Summations for L_n, R_n and M_n for f(x)=x²+1

Find the closed forms for ∑a, ∑i and ∑i² for i=1 to n and substitute into the each of the 3 Riemann sum equations.

Then all you have to do is let n = 25 and 100.

For example, L_n= ∑_i=0^n-1f(a+iΔx)Δx = ∑_i=0^n-1(3/n)[2+6i/n+9i²/n²] = (3/n)∑_i=1ⁿ[2 + 6(i-1)/n + 9(i-1)²/n²]

The sum should approach 24

Answer 2

desmos.com/calculator/ivunvbeay0