Question

A. Express the limit as a definite integral on the given interval.

n Š lim n-> Xi Ax, [1, 3] (xi +13 * 2 i=1

B. Use the form of the definition of the integral to evaluate the integral.

Answer 1

x Given sum of the limit, lim Ax, [13] M-00 (x* )* +13 From the definition of the definite integral of f from a to b we have lim Žf(x*)ax= [f(x) lim And comparing with the given sum 1 x + 13 12 lim 1-00 -Ar= 1 f(x) -1 (**)* +13 х We can see that f(x) = x +13 Finally express the limit as a definite integral on the given interval (1.3] is 3 х lim Ax = -dxl. 100 1-1 (x) +13 x + 13 1

B. Given definite integral, ((2x-x?)ox USING THE DEFINITION of the definite integral: b-a 3 Using right endpoints with n rectangles of equal width along [0.3]: Ar=- The area of rectangle using the right points such that: A = f(x)Ax, where x; = a +iAx 3i n n x = 0+ n 3i 3i n n 6i 912 n n

A=f(xar 6 972 3 A= n n 72 6 9:23 lim 100 n n in 18 2712 =lim ܐܕ M-700 i- n 100 na M-00 18 = limSi-lim 18 n(n+1) 27 n(n+1) (2n+1) =lim lim 2 on 6 18 n(n+1) 27 n(n+1)(2n+1) =lim -lim 2 6 2 1-0 n 1-on 1 =9lim 1+2 -9lim 1+ 1+ 72 12 n 2n =9-9 Since n.-→ 0 n = 0 Therefore, I(2x-xº)dx=0