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GATE PI 2019 Official Paper

__Concept:__

Trapezoidal rule is given by:

\(\mathop \smallint \limits_{\rm{a}}^{\rm{b}} {\rm{f}}\left( {\rm{x}} \right){\rm{dx}} = \frac{{\rm{h}}}{2}\left[ {{{\rm{y}}_{\rm{o}}} + {{\rm{y}}_{\rm{n}}} + 2\left( {{{\rm{y}}_1} + {{\rm{y}}_2} + {{\rm{y}}_3}{\rm{\;}} \ldots } \right)} \right]\)

\({\rm{Number\;of\;intervals\;(n)}} = \frac{{{\rm{b}}\; - \;{\rm{a}}}}{{\rm{h}}}{\rm{\;}}\)

Where b is the upper limit, a is the lower limit, h is the step size.

__Calculation:__

**Given:**

x | 0 | 0.5 | 1 |
---|---|---|---|

f( x ) = e^{-x} |
1 | 0.6065 | 0.3678 |

**By Trapezoidal rule:**

\(\int_0^1e^{-x}dx=\frac{h}{2}[y_0+y_n+2y_1]\)

\(\int_0^1e^{-x}dx=\frac{0.5}{2}[1+0.3678+(2\times0.6065)]=0.645\)

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