USE NUMERICAL APPROXIMATION TECHNIQUES TO EVALUATE DEFINITE INTEGRALS, WHERE APP
ID: 2886219 • Letter: U
Question
USE NUMERICAL APPROXIMATION TECHNIQUES TO EVALUATE DEFINITE INTEGRALS, WHERE APPROPRIATE, INCLUDING THE AREA UNDER A CURVE; USE RIEMANN SUMS, USING LEFT, RIGHT, AND MIDPOINT EVALUATION POINTS AND TRAPEZOIDAL SUMS TO APPROXIMATE DEFINITE INTEGRALS OF FUNCTIONS THROUGH ALGEBRAIC, GRAPHICAL, AND TABULAR REPRESENTATION, TABLE OF VALUES; DISCUSS ERROR IMPLICATIONS OF DIFFERENT METHODS Using 4 equal-width intervals, show that the trapezoidal rule is the average of the upper and lower sum estimates for [x2dx.Explanation / Answer
Trapezoidal sum :
?x = (4 - 0)/4
?x = 4/4
?x = 1
Integral =(1/2) * (1) * ( f(0) + 2*f(1) + 2*f(2) + 2*f(3) + f(4) )
=(1/2) * (1) * ( 0 + 2*1 + 2*4 + 2*9 + 16 )
=22
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lower Sum :
Integral = (1) * ( f(0) + f(1) + f(2) + f(3) )
=(1) * ( 0 + 1 + 4 + 9 )
=14
Upper sum :
Integral =(1) * ( f(1) + f(2) + f(3) + f(4) )
=(1) * ( 1 + 4 + 9 + 16 )
=30
Find average of Upper sum and lower Sum as
average = (Upper sum +lower Sum)/2
=(30 +14)/2
=22
===========
Clearly, Trapezoidal rule is average of Upper sum and lower Sum .
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