It appears
that you are trying to estimate the area under the curve y=2+sin(x) from 0 to class="AM">`pi` :
(1)Upper sum with n=2: The width
of the rectangles is `Delta x=pi/2` . The height
of the rectangles is the maximum the function takes on the intervals -- in both cases the
maximum occurs at `x=pi/2` .
So
the upper sum is class="AM">`(2+sin(pi/2))pi/2+(2+sin(pi/2))pi/2=3pi~~9.42`
sscr="-.785,3.93,-1,4,1.57,1,1,1.57,1,300,200,func,2+sin(x),null,0,0,black,1,none,func,3,null,0,0,0,3.145,black,1,none,func,100x-50(pi),null,0,0,1.571,1.601,black,1,none"
style="width: 300px; height: 200px; vertical-align: middle; float: none;"
type="image/svg+xml">
(2) The lower sum with n=2 : again
`Delta x=pi/2` . Now we take the lowest value the
function takes on each interval; in both cases that is 2.
Lower sum is
`(2+sin(0))pi/2+(2+sin(pi))pi/2=2pi~~6.28`
sscr="-.785,3.93,-1,4,1.57,1,1,1.57,1,300,200,func,2+sin(x),null,0,0,black,1,none,func,100x,null,0,0,0,.01,black,1,none,func,100x,null,0,0,0,.02,black,2,none,func,2,null,0,0,0,3.142,black,2,none,func,100(x-3.142),null,0,0,3.1415,3.162,black,2,none,func,100(x-1.571),null,0,0,1.571,1.591,black,2,none"
style="width: 300px; height: 200px; vertical-align: middle; float: none;"
type="image/svg+xml">
The average of the upper and lower
sums is approximately 7.85.
The actual value is approximately 8.2831853. (The
value is exactly `2+2pi` )