1. Given f(x) = 2x2 +6, find the exact area A of the region under y = f (x) on the interval [1, 3] by first computing _ f(x;)Ax, where Ax = = and then taking the limit as n co. 1=1 (Do not use integration technique; use the computational algorithms only. Show your work to get the full credit) 2. Find [ [Sx + 2 sin(x) ]dx. 0 3. (20) A particle moving along the x-axis at the origin x=0, when t=0. A time t, its velocity is v( t) = = 1 (f+ 1)2 : which is rapidly decreasing. Show that the particle will never pass the point x=1. (Hint: distance travelled at time b, is d(b) = [v(f)at ) 4. Since f(x)=1+ 2x" is an increasing function on [1,4], f(1)=3 is the minimum, and f(4)=33 is the maximum of the function f on [1.4]. The error of an integration based on a Riemann sum may be given as the difference between the upper sum and lower sum as, error(n) = _ f(M,)Ax; - _ f(m; )Ax, , where f(M;), and f(m;) are respectively, the maximum and minimum of the function, f, over the subinterval [xi-1, xi], (a) Show that error(n) =[f(4)-f(1)]Ax, when the sub-interval Ax; = Ax= (4-1) , is n of equal length. (Hint: Divide the interval [1, 4] equally into n subintervals, *j, X2, .3 Xj-1,*;,..., *n, Where x =1, and X, = 4.) (b) Compute error(100) and error(200). Is error(n) decreasing as n increases from 100 to 200