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| PAGE: 293 | SET: Exercises | PROBLEM: 1 |
Determine the area of the curve using four rectangles from 
to 
.
Number of intervals 
.
Area is the sum of areas of the 
rectangles.
Width of the each rectangle : 
.
The rectangle repeat for 
and 
.
Lower estimates :
Redraw the graph using left endpoints :
Area under the graph using left endpoints :
.
.
Observe the graph:
The value of the function at different values of 
are
, 
, 
and 
.
Substitute 
, 
, 
,
and 
in 
.
Area of the graph of the function using left endpoints is 
.
Upper estimates :
Redraw the graph using right endpoints:
Area under the graph using right endpoints :
.
.
Observe the graph:
The value of the function at different values of 
are
, , and 
.
Substitute , , , and in 
.
Area of the graph of the function using right endpoints is 
.
(b)
Determine the area of the curve using eight rectangles from to .
Observe the graph:
and 
.
Number of intervals 
.
Area is the sum of areas of the 
rectangles.
Width of the each rectangle: 
.
Determination of lower estimates:
.
.
.
Area of the graph of the function using left endpoints is .
Determination of upper estimates:
.
Area of the graph of the function using right endpoints is 
.
Lower estimate:
Graph of the function using left endpoint is
Area of the graph of the function using left endpoints is 
.
Upper estimate:
Graph of the function using right endpoint is
Area of the graph of the function using right endpoints is 
.
(b)
Area of the graph of the function using left endpoints is .
Area of the graph of the function using right endpoints is 
.
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