PYQs

1) Use five subintervals to integrate ${\int }_0^1{\displaystyle \frac{dx}{1+x^2}}$ using trapezoidal rule.

[2014, 10M]

2) Find from the following table, the area bounded by the $x-axis$ and the curve $y=f(x)$ between $x=5.34$ and $x=5.40$ using the trapezoidal rule:
$\begin{array}{|cccccccc|}\hline x& 5.34& 5.35& 5.36& 5.37& 5.38& 5.39& 5.40\\ f(x)& 1.82& 1.85& 1.86& 1.90& 1.95& 1.97& 2.00\\ \hline\end{array}$

[2007, 15M]

Starting from rest in the beginning, the speed(in Km.h) of a train at different times(in minutes) is given by the above table.

Using Simpson’s ${\displaystyle \frac{1}{3}}$rd rule, find the approximate distance travelled (in Km) in 20 minutes fro the beginning.

[2018, 10M]

2) Derive the formula ${\int }_a^{b}ydx={\displaystyle \frac{3h}{8}}[(y_0+y_n)+3(y_1+y_2+y_4+y_5+\dots +y_{n-1})+2(y_3+y_6+y_{n-3})]$. Is there any restriction on $n$? State that condition. What is the error bounded in the case of Simpson’s ${\displaystyle \frac{3}{8}}$th rule?

[2017, 20M]

3) Draw a flowchart for Simpson’s one-third rule.

[2014, 15M]

4) The velocity of a train which starts from rest is given in the following table. The time is in minutes and velocity is in km/hour.
$\begin{array}{|ccccccccccc|}\hline t& 2& 4& 6& 8& 10& 12& 14& 18& 18& 20\\ v& 16& 28.8& 40& 46.4& 51.2& 32.0& 17.6& 8& 3.2& 0\\ \hline\end{array}$
Estimate approximately the total distance run in 30 minutes by using composite Simpson’s ${\displaystyle \frac{1}{3}}$rd rule.

[2013, 15M]

5) Calculate ${\int }_2^{10}{\displaystyle \frac{dx}{1+x}}$ (up to 3 places of decimal) by dividing the range into 8 equal parts by Simpson’s ${\displaystyle \frac{1}{3}}$rd rule.

[2011, 12M]

6) A solid of revolution is formed by rotating about the $x-axis$, the area between the $x-axis$, the line $x=0$ and $x=1$ and a curve through the points with the following co-ordinates:
$\begin{array}{|cccccc|}\hline x& 0.00& 0.25& 0.50& 0.75& 1\\ y& 1& 0.9896& 0.9589& 0.9089& 0.8415\\ \hline\end{array}$

Find the volume of the solid.

[2011, 20M]

7) Find the value of the integral ${\int }_1^{\mathrm{\infty }}{\log }_{10}xdx$ by using Simpson’s ${\displaystyle \frac{1}{3}}$rd rule, correct up to 4 decimal places. Take 8 subintervals in your computation.

[2010, 20M]

8) Evaluate $I={\int }_0^1{e}^{-x^2}dx$ by the Simpson’s rule ${\int }_a^{b}f(x)dx\approx {\displaystyle \frac{\mathrm{\Delta }x}{3}}[f\left(x_0\right)+4f\left(x_1\right)+2f\left(x_2\right)+4f\left(x_3\right)+\dots +2f\left(x_{2n-2}\right)+4f\left(x_{2n-1}\right)+f\left(x_{2n}\right)]$ with $2n=10$, $\mathrm{\Delta }x=0.1$, $x_0=0$, $x_1=0.1$, $\dots$, $x_{10}=1.0$.

[2006, 12M]

9) The velocity of a particle at distance from a pint on it s path is given by the following table:
$\begin{array}{|cccccccc|}\hline S(meters)& 0& 10& 20& 30& 40& 50& 60\\ V(m/sec)& 47& 58& 64& 65& 61& 52& 38\\ \hline\end{array}$
Estimate the time taken to travel the first 60 meters using Simpson’s ${\displaystyle \frac{1}{3}}$rd rule. Compare the result with Simpson’s ${\displaystyle \frac{3}{8}}$th rule.

[2004, 12M]

10) Draw a flow chart and write a program in BASIC for Simpson’s ${\displaystyle \frac{1}{3}}$rd rule for integration ${\int }_a^b{\displaystyle \frac{1}{1+x^2}}dx$ correct to ${10}^{-6}$.

[2003, 30M]

1) Find the values of the constant $a,b,c$ such that the quadrature formula:

${\int }_o^{h}f(x)dx=h[af(o)+bf({\displaystyle \frac{h}{3}})+cf(h)]$ is exact for polynomial of as high degree as possible, and hence find the order of the truncation error.

[2018, 15M]

2) For an integral ${\int }_{-1}^{1}f(x)dx$, show that the two point Gauss quadrature rule is given by ${\int }_{-1}^{1}f(x)dx=f\left({\displaystyle \frac{1}{\sqrt{3}}}\right)+f(-{\displaystyle \frac{1}{\sqrt{3}}})$. Using this rule, estimate ${\int }_2^{4}2x{e}^{x}dx$.

[2016, 15M]

3) Use appropriate quadrature formulae out of the Trapezoidal and Simpson’s rules to numerically integrate ${\int }_0^1{\displaystyle \frac{dx}{1+x^2}}$ with $h=0.2$. Hence, obtain an approximate value of $\pi$. Justify the use of particular quadrature formula.

[2005, 12M]

4) Evaluate ${\int }_0^1{e}^{-x^2}dx$ by employing three points Gaussian quadrature formula, finding the required weights and residues. Use five decimal places for computation.

[2003, 12M]