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For the data,x : 0 1 2f(x) : 8 5 6the value of \(\displaystyle\int_0^2 [f(x)]^2 dx\) by Trapezoidal rule will be:1. 922. 753. 1234. 42 |
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Answer» Correct Answer - Option 2 : 75 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 1 2 f(x) : 8 5 6 [f(x)]2:64 25 36 From the above given data n = 2, y0 = 64, y1 = 25, y2 = 36, b = 2, a = 0 \(h = {(b-a)\over Number~ of ~intervals}={(2-0)\over 2}=1\) By using the Trapezoidal rule we get: \(\displaystyle\int_0^2 [f(x)]^2 dx={h\over2}[{y_0+y_2+2(y_1)}]={1\over 2}[64+36+2(25)]={150\over 2}=75\) |
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