Exercise 4.5 (a) [Trapezoidal and Simpson's 1/3rd rule]

EXERCISE 4.5 (a) [Trapezoidal and Simpson's 1/3^rd rule]

1. Compute the value of  using Simpson's rule and Trapezoidal rule.

Take h = 0.25.

 [Ans. 0.6931, 0.6971]

2. Evaluate  correct to two decimal places, using seven ordinates.

 [Ans. 1.3935]

3. Find the value of  using Simpson's one-third rule with h = 0.025

[Ans. 0.23108]

4. The velocity v of a particle at distance s from a point on its path is given by the table below.

Estimate the time taken to travel 60 metres by using Simpson's one-third rule.

 [Ans. 1.0635 sec]

5. When a train is moving at 30 m/sec. steam is shut off and brakes are applied. The speed of the train per second after t seconds is given by

 [Ans. 606.66 mm]

6. Evaluate  to three decimals, dividing the range of integration into 8 equal parts using Simpson's rule.

[Ans. 0.8145]

7. A solid of revolution is formed by rotating about the X-axis, the area between X-axis, x = 0, x = 1 and the curve through the points (0, 1), (0.25, 0.9896), (0.5, 0.9589), (0.75, 0.9089) and (1, 0.8415). Find the volume of solid.

[Ans. 2.819]

8. Given e⁰ = 1, e¹ = 2.72, e² = 7.39, e³ = 20.09, e⁴ = 54.60, use Simpson's rule to evaluate  Compare your result with exact value.

[Ans. 53.8733, 53.598]

9. Evaluate, dx by taking h = 0.1 using Simpson's rule.

 [Ans. 0.972]

10. Evaluate, dx by dividing the range into 8 equal parts.

[Ans. 1.299]

11. Calculate,   taking h = π / 6

 [Ans. 1.305]

12. The velocity v of a particle moving in a straight line covers a distance x in time t. They are related as follows :

Find time taken to traverse the distance of 40 units.

[Ans. 0.725 units of time]

13. Evaluate,   h = 0.05 by various methods. What is the error made in using Trapezoidal rule.

[Ans. 0.32147, 0.32149, error = 0.00002]

14. Calculate,   taking 5 ordinates by Simpson's rule.

 [Ans. 0.08409]

15. A river is 40 m wide. The depth d in metres at a distance x metres So from one bank is given by the table below:

Find the cross-section of the river by Simpson's rule.

[Ans. 190 sq.m]

16. The speed of a train at various times are given by

Find the total distance covered.

[Ans. 1.6666 km]

17. A reservoir discharging water through sluices has a depth h metre below the water surface, has a surface area A for various values of h as given below:

If t denotes time in minutes, the rate of fall of surface area is given by dh / dt = 48 √h / A. Estimate the time taken for the water level to fall from 14 to 10 metres above the sluices.

[Ans. 29 minutes nearly]

18. Evaluate  Simpson's (1/3) rule, dividing the range into four equal parts. [A.U M/J 2016 R13]

[Ans. 0.7854]