You have not provided the "form of the definition of the integral given in this theorem", nor the relevant theorem.
The integral of the expression given is x3 + 4x2.
Evaluated between 1 and 5 gives 120.
Nobody would use the Riemann sum of this expression to evaluate the integral from 1 to 5. That is the use of the Fundamental Theorem of Calculus! That evaluation would be very messy.
The Riemann sum for the first term: lim as n-->∞( 3 x 4/n)∑ [1+(4i/n)]2 and the sum would be evaluated from 1 to n. The 2nd term would be similar without the square inside the sum.
Wyzant T.
Sorry theorem given was: If f is integrable on [a,b] then, ∫ab f(x) dx = lim n --> ∞ n Σ i=1 f(xi) ∆x Where ∆x = b-a/n and xi = a + i∆x Can you show the whole solution step by step?01/30/24