Estimate the area bounded between the graph of f (x ) and the x -axis over the interval [4, 10].
1 x 4
f ( x) e
4
1) Why can't we use a single geometry formula to find the exact area?
2) What are some possible geometric shapes that we can use to estimate this area?
An upper estimate of the area A lower estimate of this area
0.25 0.25
0.23 0.23
1 x 4
0.20 f ( x) e 0.20
0.18 4 0.18
0.15 0.15
0.13 0.13
0.10 0.10
0.08 0.08
0.05 0.05
0.03 0.03
0.00 0.00
0 1 2 3 4 5 6 7 8 9 10 11 12 0 1 2 3 4 5 6 7 8 9 10 11 12
width of rectangle 6 width of rectangle 6
height of rectangle 0.0919699 height of rectangle 0.0205212
area of rectangle 0.5518192 area of rectangle 0.123127498
1) What type of units can area have?
2) Where do these values come from?
3) How can we get a better estimate without increasing the number of rectangles that we use?
0 0.2500000 4 0
0.2 0.2378074 10 0
0.4 0.2262094 4 0
0.6 0.2151770 4 0.0919699
0.8 0.2046827 10 0.0000000
1 0.1947002 10 0.0919699
1.2 0.1852046
1.4 0.1761720 4 0.0919699
1.6 0.1675800 10 0.0919699
1.8 0.1594070
2 0.1516327 4 0
2.2 0.1442375 10 0
2.4 0.1372029 4 0.0000000
2.6 0.1305114 4 0.0205212
2.8 0.1241463 4 0.0205212
3 0.1180916 10 0.0205212
3.2 0.1123322 10 0
3.4 0.1068537 10 0.0205212
3.6 0.1016424
3.8 0.0966853
4 0.0919699
4.2 0.0874844
4.4 0.0832178
4.6 0.0791592
4.8 0.0752986
5 0.0716262
5.2 0.0681329
5.4 0.0648101
5.6 0.0616492
5.8 0.0586426
6 0.0557825
6.2 0.0530620
6.4 0.0504741
6.6 0.0480125
6.8 0.0456709
7 0.0434435
7.2 0.0413247
7.4 0.0393093
7.6 0.0373922
7.8 0.0355685
8 0.0338338
8.2 0.0321837
8.4 0.0306141
8.6 0.0291210
8.8 0.0277008
9 0.0263498
9.2 0.0250647
9.4 0.0238423
9.6 0.0226795
9.8 0.0215734
10 0.0205212
10.2 0.0195204
10.4 0.0185684
10.6 0.0176628
10.8 0.0168014
11 0.0159820
11.2 0.0152025
11.4 0.0144611
11.6 0.0137558
11.8 0.0130849
12 0.0124468
An estimate using the midpoint of the interval.
0.25
0.23
1 x 4
0.20 f ( x) e
0.18 4
0.15
0.13
0.10
0.08
0.05
0.03
0.00
0 1 2 3 4 5 6 7 8 9 10 11 12
width of rectangle 6
height of rectangle 0.0434435
area of rectangle 0.2606609
1) Does the area of the rectangle give us the exact area under the graph?
2) How can we get a better estimate with these types of rectangles?
0 0.2500000 4 0
0.2 0.2378074 10 0
0.4 0.2262094 4 0
0.6 0.2151770 4 0.0434435
0.8 0.2046827 10 0.0000000
1 0.1947002 10 0.0434435
1.2 0.1852046
1.4 0.1761720 4 0.0434435
1.6 0.1675800 10 0.0434435
1.8 0.1594070
2 0.1516327 7 0
2.2 0.1442375 7 0.0434435
2.4 0.1372029
2.6 0.1305114
2.8 0.1241463
3 0.1180916
3.2 0.1123322
3.4 0.1068537
3.6 0.1016424
3.8 0.0966853
4 0.0919699
4.2 0.0874844
4.4 0.0832178
4.6 0.0791592
4.8 0.0752986
5 0.0716262
5.2 0.0681329
5.4 0.0648101
5.6 0.0616492
5.8 0.0586426
6 0.0557825
6.2 0.0530620
6.4 0.0504741
6.6 0.0480125
6.8 0.0456709
7 0.0434435
7.2 0.0413247
7.4 0.0393093
7.6 0.0373922
7.8 0.0355685
8 0.0338338
8.2 0.0321837
8.4 0.0306141
8.6 0.0291210
8.8 0.0277008
9 0.0263498
9.2 0.0250647
9.4 0.0238423
9.6 0.0226795
9.8 0.0215734
10 0.0205212
10.2 0.0195204
10.4 0.0185684
10.6 0.0176628
10.8 0.0168014
11 0.0159820
11.2 0.0152025
11.4 0.0144611
11.6 0.0137558
11.8 0.0130849
12 0.0124468
An estimate using the midpoint of the interval.
0.25
0.23
1
0.20 f ( x) e x 4
0.18 4
0.15
0.13
0.10
0.08
0.05
0.03
0.00
0 1 2 3 4 5 6 7 8 9 10 11 12
rectangle width midpoint height area
1 2 5 0.0716262 0.143252
2 2 7 0.0434435 0.086887
3 2 9 0.0263498 0.052700
0.282839 Final estimate
0 0.2500000 4 0
0.2 0.2378074 10 0
0.4 0.2262094 4 0
0.6 0.2151770 4 0.0716262
0.8 0.2046827 6 0.0716262
1 0.1947002 6 0.0000000
1.2 0.1852046
1.4 0.1761720
1.6 0.1675800 6 0.0434435
1.8 0.1594070 8 0.0434435
2 0.1516327 8 0.0000000
2.2 0.1442375 8 0.0434435
2.4 0.1372029
2.6 0.1305114
2.8 0.1241463 8 0.0263498
3 0.1180916 10 0.0263498
3.2 0.1123322 10 0.0000000
3.4 0.1068537 10 0.0263498
3.6 0.1016424
3.8 0.0966853
4 0.0919699
4.2 0.0874844
4.4 0.0832178
4.6 0.0791592
4.8 0.0752986
5 0.0716262
5.2 0.0681329
5.4 0.0648101
5.6 0.0616492
5.8 0.0586426
6 0.0557825
6.2 0.0530620
6.4 0.0504741
6.6 0.0480125
6.8 0.0456709
7 0.0434435
7.2 0.0413247
7.4 0.0393093
7.6 0.0373922
7.8 0.0355685
8 0.0338338
8.2 0.0321837
8.4 0.0306141
8.6 0.0291210
8.8 0.0277008
9 0.0263498
9.2 0.0250647
9.4 0.0238423
9.6 0.0226795
9.8 0.0215734
10 0.0205212
10.2 0.0195204
10.4 0.0185684
10.6 0.0176628
10.8 0.0168014
11 0.0159820
11.2 0.0152025
11.4 0.0144611
11.6 0.0137558
11.8 0.0130849
12 0.0124468
1
Find S30 e x 4 , [4,10]
4
A 4 Sum of the areas of the rectangles 0.285764674
B 10
N 30
x 0.20
Rectangle Width Midpoint Height Area 1) Is there an advantage to storing
1 0.2 4.1 0.08970 0.01794 A, B, and N in separate cells?
2 0.2 4.3 0.08532 0.01706
3 0.2 4.5 0.08116 0.01623
4 0.2 4.7 0.07720 0.01544
5 0.2 4.9 0.07344 0.01469 2) Is our answer the exact area?
6 0.2 5.1 0.06986 0.01397
7 0.2 5.3 0.06645 0.01329
8 0.2 5.5 0.06321 0.01264
9 0.2 5.7 0.06013 0.01203
10 0.2 5.9 0.05719 0.01144
11 0.2 6.1 0.05441 0.01088
12 0.2 6.3 0.05175 0.01035
13 0.2 6.5 0.04923 0.00985
14 0.2 6.7 0.04683 0.00937
15 0.2 6.9 0.04454 0.00891
16 0.2 7.1 0.04237 0.00847
17 0.2 7.3 0.04030 0.00806
18 0.2 7.5 0.03834 0.00767
19 0.2 7.7 0.03647 0.00729
20 0.2 7.9 0.03469 0.00694
21 0.2 8.1 0.03300 0.00660
22 0.2 8.3 0.03139 0.00628
23 0.2 8.5 0.02986 0.00597
24 0.2 8.7 0.02840 0.00568
25 0.2 8.9 0.02702 0.00540
26 0.2 9.1 0.02570 0.00514
27 0.2 9.3 0.02445 0.00489
28 0.2 9.5 0.02325 0.00465
29 0.2 9.7 0.02212 0.00442
30 0.2 9.9 0.02104 0.00421
B A This finds the width of the rectangle. We use the same width for each
x
n rectangle in the problem to make the calculations easier.
This has the same units as x.
These are the midpoints for the rectangles. They are x values.
m1 , m2 , m3 , mn The first value is found in a different way than the remaining values.
1 mk 1 mk x
m1 A x
2
f (m1 ), f (m2 ), f (mn ) These are the heights for the rectangles. They are found by plugging the
midpoints into the given function.
These are y values and have the same units as y .
n
f (m
k 1
k ) x This is the sigma form for the sum of the areas of the n rectangles.
S n ( f ,[ A, B]) This is the notation the text uses for the sum of n midpoint rectangles.
An approximation of the true area bounded by a graph over the interval [A, B] can
n
be written as S n ( f ,[ A, B]) or f (m ) x
k 1
k
If we let n or x 0 we get the exact area bounded by the graph.
n
Then we write lim f (mk ) x
x 0
k 1
We use another notation to represent the exact area. It is called the definite integral of f(x)
B
over the interval [A, B] and is written as
f ( x)dx
A
2
(ex)
0
x 1dx represents the exact area bounded between the graph of
and the x-axis over the interval [0,2].
f ( x) x 1