Question
Download Solution PDFFor the data,
x : 0 1 2
f(x) : 8 5 6
the value of \(\displaystyle\int_0^2 [f(x)]^2 dx\) by Trapezoidal rule will be:
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFConcept:
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:
\(\displaystyle\int_0^2 [f(x)]^2 dx={h\over2}[{y_0+y_2+2(y_1)}]={1\over 2}[64+36+2(25)]={150\over 2}=75\)
Last updated on May 26, 2025
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