Question

Estimate the area under the graph in the figure by using the following rules with n = 4. (Round your answers to one decimal place.) y 1 0 1 2 3 (a) Trapezoidal Rule (b) Midpoint Rule (c) Simpson's Rule

Answer 1

Let the given greeph is the greeph of function ya flee) in the interval 10,4] also n = 4 (given) Hence Ax= 40- 4 And subintervals are. Midpaints are = 0.5 [0,1], [1,2],[2,3] , [3,4] X = 9.5 Az = 25 , Xq = 3.5 ) And No=0 x 51 x₂=2 x=3 and xyrt 1 (a) By Trapezoidal Relle area under the graph flse) is. The A: f(x) dan Αχ 6 2. 1 2 [ f() +2 f121) +2 f(x2) +24(23) + frau)] [f(0) +2 ftut 2 ft2)+2+(3) + f(4)] [0+ 2x 3 + 2x5+ 2* 3+1] (from graph) (0+6 +10+&+1) = 61 12 hence A- 11.5 (fint Result) (6) By Mud point Rule 4 A= o S'f(x) dx * Ax (415)+ H #g) + 4183) + f(ay)] 1 [ flos) + f(15) + f(205)+ 43-5)] 1 [1 + 4.5+ 4.5+2] - 12 Hence A - 12 (second Result)

(c) By Simpson & Rule. A= z 1 3 flal dan [ %) ++44*,+2 f121, 3+ 4 f(x3) +41814)] [f10)+4 +10+2 f12)+44(3)+F(4)] }[ 0 +4x3+ 2x 5 + 4x3+1] = [0+ 12 + 10+12+1] - 35 3 35 11.6667 1147 (Round to one decimal peace ) (Third Result) hence An 11.7