For a Riemann sum, a TYPICAL ELEMENT is a rectangle, which looks like (e.g., with change x = 1)
14
Expression 15: 0 less than or equal to "y" less than or equal to 3 left brace, 3 less than or equal to "x" less than or equal to 4 , right brace left brace, "z" equals 0 , right brace0≤y≤33≤x≤4z=0
15
Now fill in all the typical elements as x ranges form 0 to 6, with the change in x = 1:
16
Expression 17: "J" equals left bracket, 0 , 1 , ... , 6 , right bracketJ=0,1,...,6
equals=
00
11
22
33
44
55
66
17
Expression 18: 0 less than or equal to "y" less than or equal to "J" minus 1 left brace, "J" minus 1 less than or equal to "x" less than or equal to "J" , right brace left brace, "z" equals 0 , right brace0≤y≤J−1J−1≤x≤Jz=0
18
The purple area approximates the teal area.
19
To get better and better approximation of teal area, we take the change in x smaller and small.
20
E.g., when the change in x = 0.5 we get
21
Expression 22: "K" equals left bracket, 0 , 0.5 , ... , 6 , right bracketK=0,0.5,...,6
equals=
00
0.50.5
11
1.51.5
22
2.52.5
33
3.53.5
44
4.54.5
55
5.55.5
66
22
Expression 23: 0 less than or equal to "y" less than or equal to "K" minus 0.5 left brace, "K" minus 0.5 less than or equal to "x" less than or equal to "K" , right brace left brace, "z" equals 0 , right brace0≤y≤K−0.5K−0.5≤x≤Kz=0
23
Letting the change in x go to zero gives the value of the definite integral is .....
24
Expression 25: Start integral from 0 to 6, end integral, "x" "d" "x"∫60xdx
