Question

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Let 1 = f(x) dx, where f is the function whose graph is shown. 3 (a) Use the graph to find L2, R2 and M2. L2 =

Answer 1

Given Is (t fra) dx from the graph, I = lo fcn du v Here ato, b=4 &n=2 - One- brą - thoma The intervals are (0, 2) (2, 4) → from graph f(0) = 2 I FCI) = 6 f(2)=10 f(3)=13 f(4) = 14 l (a) for left rule, The intervals are 042 - L2 - Duffo+f(2)] = 2 (2+10] 2 [12] = 24 - 24 V for Reight rule, The intervals are 2, 4 Rq = Da (f (2) + f(A)] - 2 [10+14] 2 [224] = 48 Rq = [48] u for midpoint role, The intervals are 143 Ma = Da [FCD + f(3))

= 2(6+13] - 2[19] - 38 Mg [38] Rg = [48] Mg = [38] (b) Concare down Here I-f(x) is increasing function & Log is an underestimate of I O Rq is an overestimate of I 0 Mq is an overestimate of I trapezoidal role © fa 2 | 14 T = x [(yotyz) try] - } [2+14) 420603 = [16+20] =36 10 y, yo q Here 'I' is an concave down, then Top as underestimate of I. (d) The Top lesser than I increasing order for one for La Mg, R2, Tq is Le, Tq , I, Mq , R2