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With respect to the numerical evaluation of the definite integral, $K = \int \limits_a^b \:x^2 \:dx$, where $a$ and $b$ are given, which of the following statements is/are TRUE?

  1. The value of $K$ obtained using the trapezoidal rule is always greater than or equal to the exact value of the definite integral.
  2. The value of $K$ obtained using the Simpson's rule is always equal to the exact value of the definite integral.
  1. I only
  2. II only
  3. Both I and II
  4. Neither I nor II
in Numerical Methods by Veteran (105k points) | 1k views
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1 Answer

+5 votes
Answer is C.

1 is true because when we find the true error for the given function in case of Trepoziadal rule we get E=-h^3/6 which is always less than zero. Here the error is =True value - Approx Value. Since the error is less than zero the approx value K is always greater than exact value.

2.Since the given function is a polynomial of degree 2 Simpson's will provide the exact value(Simpson's will give the exact value if the degree of the polynomial is <=3).
by Active (1.2k points)

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