search
Log In
Quick search syntax
tags tag:apple
author user:martin
title title:apple
content content:apple
exclude -tag:apple
force match +apple
views views:100
score score:10
answers answers:2
is accepted isaccepted:true
is closed isclosed:true
5 votes
1.4k views

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 1.4k views
2
Out of syllabus now

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).
Answer:

Related questions

12 votes
3 answers
1
2.1k views
The velocity $v$ (in kilometer/minute) of a motorbike which starts form rest, is given at fixed intervals of time $t$ (in minutes) as follows: t 2 4 6 8 10 12 14 16 18 20 v 10 18 25 29 32 20 11 5 2 0 The approximate distance (in kilometers) rounded to two places of decimals covered in 20 minutes using Simpson's $1/3^{rd}$ rule is ________.
asked Feb 16, 2015 in Numerical Methods jothee 2.1k views
1 vote
1 answer
2
448 views
If the trapezoidal method is used to evaluate the integral obtained $\int_{0}^{1} x^2dx$, then the value obtained is always > (1/3) is always < (1/3) is always = (1/3) may be greater or lesser than (1/3)
asked Nov 3, 2014 in Numerical Methods Ishrat Jahan 448 views
0 votes
1 answer
3
572 views
The trapezoidal method is used to evaluate the numerical value of $\int_{0}^{1}e^x dx$. Consider the following values for the step size h. 10-2 10-3 10-4 10-5 For which of these values of the step size h, is the computed value guaranteed to be correct to seven decimal places. Assume that there are no round-off errors in the computation. iv only iii and iv only ii, iii and iv only i, ii, iii and iv
asked Oct 30, 2014 in Numerical Methods Ishrat Jahan 572 views
0 votes
1 answer
4
516 views
The trapezoidal method to numerically obtain $\int_a^b f(x) dx$ has an error E bounded by $\frac{b-a}{12} h^2 \max f&rsquo;&rsquo;(x), x \in [a, b]$ where $h$ is the width of the trapezoids. The minimum number of trapezoids guaranteed to ensure $E \leq 10^{-4}$ in computing $\ln 7$ using $f=\frac{1}{x}$ is 60 100 600 10000
asked Sep 29, 2014 in Numerical Methods Kathleen 516 views
...