GATE IT 2005 | Question: 2

Question

If the trapezoidal method is used to evaluate the integral obtained $\int_{0}^{1} x^2dx$, then the value obtained

• A. is always > (1/3)
• B. is always < (1/3)
• C. is always = (1/3)
• D. may be greater or lesser than (1/3)

Answer 1

Answer is (A)

The approximate value calculated by the trapezoidal method is always greater than the actual value. This can be supported by the argument that value of ERROR= EXACT VALUE- APPROXIMATED VALUE  comes out to be NEGATIVE which shows that approximated value is greater.

Comments

Not for all functions rt?

http://tutorial.math.lamar.edu/Classes/CalcII/ApproximatingDefIntegrals.aspx

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But the given function is strictly decreasing from 0 1o 1 .

can we use this term :"non-monotonically" ??

A function is either "monotonically increasing or strictly increasing or None "

It overestimates the value for all the non-linear curves

http://en.wikipedia.org/wiki/Trapezoidal_rule

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yes. I meant that the answer is specific to this question and not applicable for all functions. I don't know exactly which all functions it can be true but I guess it won't be true for any functions which is not monotonically increasing or decreasing.

In the wiki page it is given- overestimate for all concave curves and underestimate for all convex curves.