The graph of a function f on the interval [0, 8] is shown in the figure. Compute the Riemann sum for fon [0, 8] using four subintervals of equal length and choosing the evaluation points to be (a) the leftendpoints, (b) the right endpoints, and (c) the midpoints of the subintervals.y31+++2345617 8 x-1-2-3 +-4-5

a) The left Riemann sum for the function fx over the interval a,b using n subintervals is given by, ∑k=0n-1fxk∆x=∆xfx0+fx1+fx2+…+fxk-1 where ∆x=b-an. Here, ∆x=b-an=8-04=2 Divide the interval 0,8 into subintervals of length of 2 as 0,2,2,4,4,6,6,8.The endpoints are, x0=0x1=2x2=4x3=6x4=8 Then, the Riemann sum using left endpoints is, ∑k=03fxk∆x=∆xfx0+fx1+fx2+fx3=2f0+f2+f4+f6=23+3+3+0=29=18 Hence, the Riemann sum using left endpoints is 18.b) The Riemann sum using right endpoints is, ∑k=1nfxk∆x=∆xfx1+fx2+⋯+fxn∑k=14fxk∆x=∆xfx1+fx2+fx3+fx4=2f2+f4+f6+f8=23+3+0-1=25=10 Hence, the Riemann sum using right endpoints is 10.c) The Riemann sum using midpoints of the subintervals is, ∑k=0n-1fxk+xk+12∆x=∆xfx0+x12+fx1+x22+⋯+fxn-1+xn2∑k=03fxk+xk+12∆x=∆xfx0+x12+fx1+x22+fx2+x32+fx3+x42=2f1+f3+f5+f7=24+2+1-2=25=10 Hence, the Riemann sum using midpoints of the subintervals is 10.