With respect to the numerical evaluation of the definite integral, \(K = \int…
2014
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.
- A.
I only
- B.
II only
- C.
Both I and II
- D.
Neither I nor II
Attempted by 18 students.
Show answer & explanation
Correct answer: C
Answer: Only the second statement is true.
Reasoning:
Trapezoidal rule: The function x^2 has second derivative f''(x)=2>0, so it is convex. For convex functions the trapezoidal approximation is less than or equal to the true integral, so the statement that the trapezoidal rule is always greater than or equal to the exact value is false.
Simpson's rule: Simpson's rule is exact for all polynomials of degree at most 3. Since x^2 is a polynomial of degree 2, Simpson's rule yields the exact value of the definite integral, so the second statement is true.
Conclusion: Only the second statement is true, so the correct choice is the option that indicates only the second statement holds.