Need the answers

2 4. This question asks you to find an integral using Riemann sums. Recall from lectures, that a definite integral / f(x) dx can be defined as n-too lim In = / f(x) dx = lim Un where Un and In are upper and lower Riemann sums respectively. (a) Sketch a diagram showing the rectangles corresponding to upper and lower Rie- mann sums, Un and Ly on n equal subintervals for the integral Include the ith subinterval in your diagram. b) Write down expressions for Un and In using sigma notation. (c) The sum of the squares of the first n positive integers is given by the following formula: 12 + 22 + 32 + 42 + 52 + ... + ( n - 1)2 + n2 = \\ k2 _ n(n + 1)(2n+ 1) 6 Use this fact to show that Un = (n + 1)(2n + 1) and (harder) that Ln = (n -1)(2n - 1) 6n2 6n2 (d) Find lim Un and lim In. Verify, using the Fundamental Theorem of Calcu- n-+0o n- 400 lus (that is, conventional methods of integration), that these limits are equal to x2 dx