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| PAGE: 540 | SET: Exercises | PROBLEM: 2 |
(a) The left, right, Trapezoidal, and Midpoint Rule approximations are 
and 
respectively.
observe the graph,
The function is decreasing and concave upwards.
For left end method:
For decreasing function left end method is generally gives over estimates since the upper right corners of the rectangles are outside the region.
Therefore the largest estimation will be 
.
For right end method:
For decreasing function right end method is generally gives under estimates since the lower left corners of the rectangles are inside the region.
Therefore the smallest estimation will be 
.
For trapezoidal method:
Trapezoidal rule gives the top sides of the trapezoids connecting two points on the curve. since it is concave up then there will be some extra space outside the region, so this will tends to be a slight overestimate.
For mid point method:
Midpoint rule will have area inside the region only.
That means it will contain the area inside the blue region not added to that outside white region.
So the mid point rule 
and the trapezoidal estimations is 
.
Therefore,
, 
, 
and 
.
(b)
From the above description,
Left end and right end methods are over and under estimates respectively.
Trapezoidal estimation is also slight over estimate and midpoint is slightly under estimate.
Therefore true value of 
lies between trapezoidal and midpoint estimations.
(a)
, 
, 
and 
.
(b) True value of 
lies between trapezoidal and midpoint estimations.
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