# GATE2014-3-46

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
2
Out of syllabus now

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).

## Related questions

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 ________.
1 vote
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)
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
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