Out of syllabus now

5 votes

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?

- The value of $K$ obtained using the trapezoidal rule is always greater than or equal to the exact value of the definite integral.
- The value of $K$ obtained using the Simpson's rule is always equal to the exact value of the definite integral.

- I only
- II only
- Both I and II
- Neither I nor II

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).

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).