BCA Semester 4 – Numerical Methods – Unit 3: Numerical Differentiation and Integration (5Hrs.)

Question 1

Write an algorithm to compute integration using Simpson's 1/3 rule. Evaluate integration of \[ \quad \quad \quad \quad ∫^30(3x2+2x−5)dx \] using Simpson's 1/3 rule.

Marks: 5

Year: 2021 Final TU FOHSS

(A) Algorithm: Simpson’s 1/3 Rule Step 1: Read the limits of integration \(a, b\) and the number of subintervals \(n\) (Note: \(n\) must be even) Step 2: Compute step size \[ h = \frac{b-a}{n} \

Question 2

Using the trapezium rule, evaluate the integral \[ I=∫^01\frac{dx}{x2+6x+10} \] with 2 and 4 sub intervals. Compare with the exact solution. Comment on the magnitudes of the errors obtained.

Marks: 5

Year: 2022 Final TU FOHSS

Given: \[ I = \int0^1 \frac{dx}{x^2 + 6x + 10} \] --- Step 1: Exact Solution The integrand can be rewritten: \[ x^2 + 6x + 10 = (x+3)^2 + 1 \] \[ I = \int0^1 \frac{dx}{(x+3)^2 + 1} = \left. \arctan(x

Question 3

For solving a linear system, compare Gauss elimination method and Gauss Jordan method. Why Gauss Seidel method is better than Jacobi's iterative method?

Marks: 5

Year: 2022 Final TU FOHSS

Given: \[ I = \int0^1 \frac{dx}{x^2 + 6x + 10} \] --- Step 1: Exact Solution The integrand can be rewritten: \[ x^2 + 6x + 10 = (x+3)^2 + 1 \] \[ I = \int0^1 \frac{dx}{(x+3)^2 + 1} = \left. \arctan(x

Question 4

Using Euler's method find \(y(0.2)\) from \[ \frac{dy}{dx}=x+y,\quad y(0)=1\] with h = 0.

Marks: 5

Year: 2022 Final TU FOHSS

Given: \[ I = \int0^1 \frac{dx}{x^2 + 6x + 10} \] --- Step 1: Exact Solution The integrand can be rewritten: \[ x^2 + 6x + 10 = (x+3)^2 + 1 \] \[ I = \int0^1 \frac{dx}{(x+3)^2 + 1} = \left. \arctan(x

Question 5

Write an algorithm and program to calculate integration using Trapezoidal rule.

Marks: 5

Year: 2023 Final TU FOHSS

1. Algorithm (Trapezoidal Rule) Goal: Approximate \(\inta^b f(x) dx\) using \(n\) subintervals. Steps: 1. Input: Function \(f(x)\), limits \(a\) and \(b\), number of subintervals \(n\). 2. Compute s

Question 6

Use the Romberg method to get an improved estimate of the integral from x = 1.8 to x = 3.4 from the data in the table with h = 0.4. X: 1.6 | 1.8 |2.0|2.2|2.4|2.6|2.8|3.0|3.2|3.4|3.6|3.8| -|-|-|-|-|-|-

Marks: 5

Year: 2024 Final TU FOHSS

1. Problem Statement Estimate the integral \[ I = \int{1.8}^{3.4} f(x)\,dx \] using Romberg method with step size \(h = 0.4\), based on the table of values: | X | 1.6 | 1.8 | 2.0 | 2.2 | 2.4 | 2.6

Question 7

Write a program to compute integral \(∫₀^{π/2} \) \(√sinx \space dx\) Simpson's \(1/3\) rule.

Marks: 5

Year: 2024 Final TU FOHSS

1. Problem Statement Compute the integral: \[ I = \int{0}^{\pi/2} \sqrt{\sin x} \, dx \] using Simpson's 1/3 Rule. --- 2. Simpson's 1/3 Rule For \(n\) intervals (even), step size \(h = \frac{b-a}{n}\

X: 1.6	1.8	2.0	2.2	2.4	2.6	2.8	3.0	3.2	3.4	3.6	3.8
Y: 4.95	6.05	7.38	9.02	11.02	13.46	16.44	0.05	24.53	29.96	36.59	44.70
:	39	5	3	4	5	6	3	4	8	1