(a) Using the trapezium rule with seven ordinates, evaluate \(\int_{0}^{3} \frac{\mathrm d x}{x^{2} + 1}\), correct to two decimal places. (b) Using matrix ...
Question 1 Report
(a) Using the trapezium rule with seven ordinates, evaluate \(\int_{0}^{3} \frac{\mathrm d x}{x^{2} + 1}\), correct to two decimal places.
(b) Using matrix method, solve \(-2x + y = 3; - x + 4y = 1\).
Answer Details
(a) Trapezium rule, seven ordinates, \(\displaystyle\int_{0}^{3}\frac{dx}{x^{2}+1}\).
Seven ordinates means six strips, so \(h=\dfrac{3-0}{6}=0.5\). With \(y=\dfrac{1}{x^{2}+1}\):
| x | 0 | 0.5 | 1 | 1.5 | 2 | 2.5 | 3 |
|---|---|---|---|---|---|---|---|
| y | 1.0000 | 0.8000 | 0.5000 | 0.3077 | 0.2000 | 0.1379 | 0.1000 |
(b) Matrix method: \(-2x+y=3,\ -x+4y=1\).
\[\begin{pmatrix}-2&1\\-1&4\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix}=\begin{pmatrix}3\\1\end{pmatrix}\]Determinant \(=(-2)(4)-(1)(-1)=-8+1=-7\). Inverse:
\[\begin{pmatrix}x\\y\end{pmatrix}=\frac{1}{-7}\begin{pmatrix}4&-1\\1&-2\end{pmatrix}\begin{pmatrix}3\\1\end{pmatrix}=\frac{1}{-7}\begin{pmatrix}12-1\\3-2\end{pmatrix}=\frac{1}{-7}\begin{pmatrix}11\\1\end{pmatrix}\]\[x=-\frac{11}{7},\qquad y=-\frac{1}{7}\]Download The App On Google Playstore
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