PYQs

1) Use five subintervals to integrate \(\int_{0}^{1} \dfrac{d x}{1+x^{2}}\) 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:
\(\begin{array}{|c|c|c|c|c|c|c|c|}\hline x & {5.34} & {5.35} & {5.36} & {5.37} & {5.38} & {5.39} & {5.40} \\ \hline f(x) & {1.82} & {1.85} & {1.86} & {1.90} & {1.95} & {1.97} & {2.00} \\ \hline\end{array}\)

[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 \(\dfrac{1}{3}\)rd rule, find the approximate distance travelled (in Km) in 20 minutes fro the beginning.

[2018, 10M]

2) Derive the formula \(\int_{a}^{b} y d x=\dfrac{3 h}{8}\left[\left(y_{0}+y_{n}\right)+3\left(y_{1}+y_{2}+y_{4}+y_{5}+\ldots+y_{n-1}\right)+2\left(y_{3}+y_{6}+y_{n-3}\right)\right]\). Is there any restriction on \(n\)? State that condition. What is the error bounded in the case of Simpson’s \(\dfrac{3}{8}\)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.
\(\begin{array}{|c|c|c|c|c|c|c|c|c|c|}\hline t & {2} & {4} & {6} & {8} & {10} & {12} & {14} & {18} & {18} & {20} \\ \hline v & {16} & {28.8} & {40} & {46.4} & {51.2} & {32.0} & {17.6} & {8} & {3.2} & {0} \\ \hline\end{array}\)
Estimate approximately the total distance run in 30 minutes by using composite Simpson’s \(\dfrac{1}{3}\)rd rule.

[2013, 15M]

5) Calculate \(\int_{2}^{10} \dfrac{d x}{1+x}\) (up to 3 places of decimal) by dividing the range into 8 equal parts by Simpson’s \(\dfrac{1}{3}\)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:
\(\begin{array}{|c|c|c|c|c|c|}\hline x & {0.00} & {0.25} & {0.50} & {0.75} & {1} \\ \hline y & {1} & {0.9896} & {0.9589} & {0.9089} & {0.8415} \\ \hline\end{array}\)

Find the volume of the solid.

[2011, 20M]

7) Find the value of the integral \(\int_{1}^{\infty} \log _{10} x d x\) by using Simpson’s \(\dfrac{1}{3}\)rd rule, correct up to 4 decimal places. Take 8 subintervals in your computation.

[2010, 20M]

8) Evaluate \(I=\int_{0}^{1} e^{-x^{2}} d x\) by the Simpson’s rule \(\int_{a}^{b} f(x) d x \approx \dfrac{\Delta x}{3}\left[f\left(x_{0}\right)+4 f\left(x_{1}\right)+2 f\left(x_{2}\right)+4 f\left(x_{3}\right)+\ldots +2 f\left(x_{2 n-2}\right)+4 f\left(x_{2 n-1}\right)+f\left(x_{2 n}\right) \right]\) with \(2 n=10\), \(\Delta x=0.1\), \(x_{0}=0\), \(x_{1}=0.1\), \(\ldots\), \(x_{10}=1.0\).

[2006, 12M]

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

[2004, 12M]

10) Draw a flow chart and write a program in BASIC for Simpson’s \(\dfrac{1}{3}\)rd rule for integration \(\int_{a}^{b} \dfrac{1}{1+x^{2}} d x\) correct to \(10^{-6}\).

[2003, 30M]

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

\(\int^h_o f(x) dx=h[af(o)+bf(\dfrac{h}{3})+cf(h)]\) 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 \(\int_{-1}^{1} f(x) d x\), show that the two point Gauss quadrature rule is given by \(\int_{-1}^{1} f(x) d x=f\left(\dfrac{1}{\sqrt{3}}\right)+f\left(-\dfrac{1}{\sqrt{3}}\right)\). Using this rule, estimate \(\int_{2}^{4} 2 x e^{x} d x\).

[2016, 15M]

3) Use appropriate quadrature formulae out of the Trapezoidal and Simpson’s rules to numerically integrate \(\int_{0}^{1} \dfrac{d x}{1+x^{2}}\) with \(h=0.2\). Hence, obtain an approximate value of \(\pi\). Justify the use of particular quadrature formula.

[2005, 12M]

4) Evaluate \(\int_{0}^{1} e^{-x^{2}} d x\) by employing three points Gaussian quadrature formula, finding the required weights and residues. Use five decimal places for computation.

[2003, 12M]