IGNOU BMTE 144 2026 - ) Evaluate by using trapezoidal rule with and&nb

Question

) Evaluate $\int_{0}^{1} \frac{1}{1 + x^2} dx$ by using trapezoidal rule with $h = 0.5$ and $h = 0.25$ . Use Romber's method to find the best value of $\pi$ .

) Estimate the eigenvalues of the matrix

$$\begin{bmatrix} 1 & -1 & 2 \\ -1 & 1 & 2 \\ 2 & 2 & -2 \end{bmatrix}$

using the Gerschgorin bounds.

09 Jan 2026

Answer :

Word Count : 361

First, we evaluate the integral [ \int_0^1 \frac{1}{1+x^2} , dx ] using the trapezoidal rule. Step 1: Trapezoidal Rule with (h = 0.5) The formula: [ T = \frac{h}{2} \big[f(x_0) + 2f(x_1) + f(x_2)\big] ] Divide [0,1] with (h=0.5): (x_0=0, x_1=0.5, x_2=1) [ f(0) = \frac{1}{1+0^2} = 1 ] [ f(0.5) = \frac{1}{1+0.5^2} = \frac{1}{1.25} = 0.8 ] [ f(1) = \frac{1}{1+1^2} = \frac{1}{2} = 0.5 ] [ T_{0.5} = \frac{0.5}{2} \big[1 + 2(0.8) + 0.5\big] = 0.25 \cdot (1 + 1.6 + 0.5) = 0.25 \cdot 3.1 = 0.775 ] Step 2: Trapezoidal Rule with (h = 0.25) Divide [0,1] with (h=0.25): (x_0=0, x_1=0.25, x_2=0.5, x_3=0.75, x_4=1) [ f(0.25) = \frac{1}{1+0.0625}

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