limits and intergral

Question

Given  , express the limit as   as a definite integral, that is provide ,  and  in the expression . 

Answer 1

The numbers 3 + (i-1)4/n yield 3, 3 + 4/n, 3 + 8/n,..., 3 + 4(n-1)/n as i varies from 1 to n. These n points partition the closed interval [3,7] into n subintervals of length 4/n, so the sum you have above is a left Riemann sum of f(x) = 5x² - 7x⁴ over the interval [3,7]. Therefore, the limit of the sum is the definite integral ∫_3^7 (5x² - 7x⁴) dx. In particular, a = 3, b = 7, and f(x) = 5x² - 7x⁴.