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 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 In on n equal subintervals for the integral 2 da. 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 + 2 k2 _ n(n+ 1)(2n + 1) 6 Use this fact to show that Un = (n + 1)(2n + 1) and (harder) that L. = (n -1)(2n - 1) 6n2 6n2 (d) Find lim Un and lim Ln. Verify, using the Fundamental Theorem of Calcu- n -too lus (that is, conventional methods of integration), that these limits are equal to x2 dx