Question

3. Consider f(x - 2) de; n = 4. Complete the following steps (a) Calculate Ax and the grid points d'o, ... assuming a regular partition (b) Calculate the right Riemann sums for the given value of n. (c) Determine the right Riemann sum underestimates or overestimates the value of the definite integral.

Answer 1

Ano Gicun f(a) = 3C2-2 to find y (x2) 2) dsi using neu right reimann sum •The right reimann sum is given by by Flx) dx = 000 (f(x,) +Flxı) +F(X3)+ - --t Flxn-1) + f (an) where sol=b-a n=4 sa = 2-0=1 4 2 a 3x2= } ) x 3 = 3 / 3 2 @ forgiven a= b=2 Divide the interval (0,2) into n=4 subintervals of length ox= 5 = [0, 1] , [4, 1], [13] , [2, 2]=b i 2,51 , Xy= 2 (Taring rightend points, for intervals b Now just evaluate the function at rightendpoints F(X) = f4) =(4)2-2 7 F12/2) = f(1) = 12-2 = -1 f(x3) Fl})= ()2-2 = 9 -2 = Flaxy) - F(2) = 22-2-4-2 = 2 Finally just sumub these values and mutiply by dx = { 1 4 2 2 -7 = -1.75 yo -0.25

Right Reimann sum 3 el (-1.75 -110.25+2) = -0.25 2 © Now S(ol ²-2) dx = 23.4 - 4 - 8-12 3-4 3 3 3 = -1.33 we see that exact value is less than right reimannisum so the right rcimann sum overestimates the value of definite integral