GATE Mechanical Engineering (ME) 2020 Shift 2 Solved Paper

$\mathit{y}=\underset{0}{\overset{\mathit{\pi }/2}{\int }}\left(8+4\cos \mathit{x}\right)\mathit{d}\mathit{x}$
Absolute percentage error in numerical evaluation with the trapezoidal rule, using only the end points:-

$={\left\{8\mathit{x}+4\sin \mathit{x}\right\}}_0^{\frac{\mathit{\pi }}{2}}$
$\left\{4\mathit{\pi }+4\sin \frac{\mathit{\pi }}{2}-8\times 0-4\times \sin 0\right\}$
${\mathit{y}}_{\mathrm{actual}}=4\mathit{\Pi }+4$
Using Trapezoidal Rule:

$=\frac{\mathit{\pi }}{4}\left[\left(8+4\cos 0\right)+\left(8+4\cos \frac{\mathit{\pi }}{2}\right)\right]$
$=\frac{\mathit{\pi }}{4}\left[\left(8+4\right)+\left(8+0\right)\right]$
$=5\mathit{n}$
Error in numerical evaluation
$=\frac{\left(4\mathit{\pi }+4\right)-5\mathit{\pi }}{\left(4\mathit{\pi }+4\right)}\times 100$
$=5.182\%$