Question

OCT/NOV2017 1 Question 4 (a) For the two-term Gaussian formula for the mtegral between-1 and 1, we have the points tt -0 5773 and tz 0 5773, as well as the weights wi 1 and u21 (8) Derive the two-point Gaussian quadıatuie foimula (2) (b) What is the degrec of the appioxmating poly nomial on which the Simpson s based (c) Find an approxiination foi the integi al (7) by means of the coinposSite Sunpson s rule for 3 submtervals (or panels) 17] TOTAL [90]

Answer 1

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aiti ta,t ayt tat 2 2 Solvi the frat fnd tat fow epnation& we 21 t--t 0.5733 ay ren t-0-577, tz- 0s773, he desree 4 the appmat pelynominl 2 tchug 2. pay-3

The two -point Gaussian formula for the integral between -1 and 1, with the points t +0.5773, and weights Wi = 1,w2 -0.5773, and _ 1 is given by t2 _ 1 1f(t,) 1f (t2) f(t)dt -1 1f(-0.5773) f(0.5773) _ b) Degree of approximating polynomial in Simpson's 1/3rd rule is 2. y-(x+1)A3 4v 2y 0 1 32 2 27 54 64 256 4 125 250 216 5 864 343 Total 1152 304 Here the three subintervals are [0, 21, [2, 4], [4, 6], in each subintervals a middle point is 0, 1, 2,3,,5,6. And h = 1. taken, thus the points are x = 6 h (x1)3 dx yo + 4(y ys) 2(y2 4) y] 3 1 [1800] [11152 304 +3431 600 _

c) Note that Simpson's 1/3rd rule is applied for even number of subintervals. But in the question it is given 3 subintevals.

For 6 subintervals the rule can be applied and is called composite Simpson 1/3rd rule.