PYQs

1) Use five subintervals to integrate ∫10dx1+x2 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:
x5.345.355.365.375.385.395.40f(x)1.821.851.861.901.951.972.00

[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 13rd rule, find the approximate distance travelled (in Km) in 20 minutes fro the beginning.

[2018, 10M]

2) Derive the formula ∫baydx=3h8[(y0+yn)+3(y1+y2+y4+y5+…+yn−1)+2(y3+y6+yn−3)]. Is there any restriction on n? State that condition. What is the error bounded in the case of Simpson’s 38th 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.
t2468101214181820v1628.84046.451.232.017.683.20
Estimate approximately the total distance run in 30 minutes by using composite Simpson’s 13rd rule.

[2013, 15M]

5) Calculate ∫102dx1+x (up to 3 places of decimal) by dividing the range into 8 equal parts by Simpson’s 13rd 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:
x0.000.250.500.751y10.98960.95890.90890.8415

Find the volume of the solid.

[2011, 20M]

7) Find the value of the integral ∫∞1log10xdx by using Simpson’s 13rd rule, correct up to 4 decimal places. Take 8 subintervals in your computation.

[2010, 20M]

8) Evaluate I=∫10e−x2dx by the Simpson’s rule ∫baf(x)dx≈Δx3[f(x0)+4f(x1)+2f(x2)+4f(x3)+…+2f(x2n−2)+4f(x2n−1)+f(x2n)] with 2n=10, Δx=0.1, x0=0, x1=0.1, …, x10=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:
S(meters)0102030405060V(m/sec)47586465615238
Estimate the time taken to travel the first 60 meters using Simpson’s 13rd rule. Compare the result with Simpson’s 38th rule.

[2004, 12M]

10) Draw a flow chart and write a program in BASIC for Simpson’s 13rd rule for integration ∫ba11+x2dx correct to 10−6.

[2003, 30M]

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

∫hof(x)dx=h[af(o)+bf(h3)+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 ∫1−1f(x)dx, show that the two point Gauss quadrature rule is given by ∫1−1f(x)dx=f(1√3)+f(−1√3). Using this rule, estimate ∫422xexdx.

[2016, 15M]

3) Use appropriate quadrature formulae out of the Trapezoidal and Simpson’s rules to numerically integrate ∫10dx1+x2 with h=0.2. Hence, obtain an approximate value of π. Justify the use of particular quadrature formula.

[2005, 12M]

4) Evaluate ∫10e−x2dx by employing three points Gaussian quadrature formula, finding the required weights and residues. Use five decimal places for computation.

[2003, 12M]