Fundamental Theorems of Calculus

Question

1.  Consider the function f:[3,6]→Real defined by f(x)=x².
2. If the domain interval of function f is subdivided into n subintervals of equal width, then the right endpoint of each subinterval is of the form 3+ (3i/n). Explain.
3. Find an explicit expression for an approximation of ∫f(x) dx, from 3 to 6, using a right sum with n subintervals of equal width, and then take a limit as n→∞ to compute an exact evaluation of ∫f(x)dx from 3 to 6.

Answer 1

a) Δx = (6-3)/n = 3/n

3 is the lower bound and 3i/n is iΔx

So each right end point is 3+3i/n

b)

I ≈ ∑ⁿ_i=1f(x) Δx = 3/n∑ⁿ_i=1 (3+3i/n)² = 3/n∑ⁿ_i=1 (9 +18i/n + 9i²/n²)

= 27/n(n + 2(n+1)n/(2n) + n(n+1)(2n+1)/(6n²))

lim n→∞ I = 27 + 27 + 9 = 63