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Evaluate ∫ 6 0 d x 1 + 3 x by using 1} Trapezoidal 2} Simpsons (1/3) rd. and 3} Simpsons (3/8) Th rule. - Applied Mathematics 2

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Evaluate `int_0^6 dx/(1+3x)`by using 1} Trapezoidal 2} Simpsons (1/3) rd. and 3} Simpsons (3/8) Th rule. 

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Solution

0 1 2 3 4 5 6
Y 1 0.25 0.1428 0.1 0.0769 0.0625 0.0526
Ordinate `y_0` `y_1` `y_2` `y_3` `y_4` `y_5` `y_6`

Trapezoidal Rule: `I=h/2(x+2R)` 

X = Sum of extreme value = 1 + 0.0526 = 1.0526 

R = Sum of Remaining values = 0.25 + 0.1428 + 0.1 + 0.0769 + 0.0625 = 0.6322 

`I=1/2(1.0526+2(0.6322))`

`I=1.1585`  

`"Simpsons (1/3) rd rule" ` 

`I=h/3 (X+2E+40)` 

X = Sum of Extreme values =1 + 0.0526 = 1.0526
E = Sum of even ordinates = 0.1428 + 0.0769 = 0.2197
O = Sum of odd ordinates = 0.25 + 0.1 + 0.0625 = 0.4125 

`I=1/3 (1.0526+2(0.2197)+4(0.4125)) ` 

`I=0.5616`

 

Concept: Simple Application of Differential Equation of First Order and First Degree to Electrical and Mechanical Engineering Problem
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