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| PAGE: 265 | SET: Exercises | PROBLEM: 69 |
The functions is 
and the equations 
.
(a)
Graph of the lower sum:
Find lower sum.
The function 
is increasing over the interval 
.
Number of subintervals are 
.
Width 
.
The left end point is 
.
.
The lower sum is 
.
The lower sum is 
sq-units.
(b)
Graph of the upper sum:
Find upper sum.
The function is increasing over the interval .
The right end point is 
.
.
Upper sum is 
.
The upper sum is 
sq-units.
(c)
Graph of the sum using mid point rule:
Using mid point theorem:
The area is 
.
Consider 
.
Where 
, 
and 
.
Substitute 
in .
.
Substitute 
values in 
.
.
Using mid point theorem:
Area =
.
(d)
Find the formulas for lower sum, upper sum and middle point rule when number of subintervals are 
.
Find lower sum.
The function is increasing over the interval .
The number of subintervals are .
The width 
.
The left end point is .
.
The lower sum is .
Find upper sum.
The right end point is .
Substitute 
in .
.
Upper sum is .
.
Find the area by using mid point rule.
Using mid point theorem:
The area is 
.
Consider .
Where is number of subintervals, and .
Substitute in .
.
Substitute values in .
.
.
(e)
Complete the table by using,
, 
and 
.
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(f)
Since is increasing function, is always increasing and is always decreasing.
(a) Graph of the lower sum:
sq-units.
(b)
Graph of the upper sum:
sq-units.
(c)
Graph of the sum using mid point rule:
sq-units.
(d)
, and .
(e)
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(f)
Since is increasing function, is always increasing and is always decreasing.
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