Question on Integrating using Simpson’s Rule

Here is a Multi choice question to solve by Integration. and Simpson’s Rule. You can find similar set of questions at TutorVista Blogs.

Topic : Integration.

Calculation in the solution will help to understand the Simpson Rule and to identify correct choice and get reasons for, why remaining choices are incorrect.

Question : Using Simpson’s Rule with n=4, the approximate value of ₀∫²5x⁴dx is
a. 12.5
b. 386/12
c. 385/12
d. 12

Solution :
Choice c is correct.

Partition [0, 2] into four sub-intervals and evaluate y=5x⁴ at the partition points

x ------- y = 5x⁴
0 ------- 0
1/2 ----- 5/16
1 ------- 5
3/2 ----- 405/16
2 ------- 80

By Simpson Rule with n=4 and h=1/2

S = h/3 (y₀ + 4y₁ + 2y₂ + 4y₃ + y₄)

= 1/6[(0)+4(5/10)+2(5)+4(405/16)+80]

= 385/12

Choice a is incorrect as the value of S comes out to be 385/12.

Choice b is incorrect as the value miscalculating the value as 386/12 instead of 385/12

Choice d is incorrect as the exact value can be verified to be 32 which is far away from 12.

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