5.2 Relevance and Accuracy of Proposed Methods
5.2.2 Performance of One-Cell Method at Multiple Interface-cells
Acrucial observation to make about this processis that as the bases of the rectangles become smaller and smaller,the sum of the areas of the rectangles appears to approach the area of R.This suggests that the area ofRshould be defined as the limit (in a sense to be clarified later)of the sum of the areas of inscribed or circumscribed rectangles.Our definition of area will be based on this idea.
Our assertion thus far about the area ofRhas rested on the following three basic properties we expect area to possess:
(1) The Rectangle Properties: The area of a rectangle is the product of its base and height.
(This property is treated as the definition of area of a rectangle.)
(2) The Addition Properties: The area of a region composed of several smaller regions that overlap in at most a line segment is the sum of the areas of the smaller regions.(4)
(3) The Comparison Property: The area of a region that contains a second region is at least as large as the area of the second region.(5)
It is important to understand where each of these properties was employed in the preceding discussion.They will play a major role in the definition of area to be discussed.
That there is a definite meaning in speaking of the area of this region is an assumption inspired by intuition. We denote the area of this region byFbaand call it thedefinite integral of the function f(x) between the limits a and b. When we actually seek to assign a numerical value to this area, we find that we are, in general, unable to measure areas of such regions with curved boundaries.
However, there is a way out. We adopt a method (based on Archimedes’ method of exhaustion), which as you will see applies to more complex regions.The method involves the summation of the areas of rectangles.
The whole process is explained below through a very simple example. Consider a right triangle formed by the linesy¼f(x)¼2x, y¼0 (thex-axis), andx¼1, as shown in the Figure 5.8.
Letb¼length of the base andh¼length of the height, then, from geometry, the areaAof the triangle is
A¼1
2bh¼1
2 ð1Þ ð2Þ ¼1 square unit
Next, we can also determine the area of this regionby another method, as suggested in the discussion above. [We have chosen a simple functionf(x)¼2x, to explain the method easily and simplify calculations for checking the results.]
Let us divide the interval [0,1] on thex-axis intofour subintervals of equal lengthDx. This is done by equally spaced points x0¼0, x1¼1/4, x2¼2/4, x3¼3/4, and x4¼4/4¼1 (see Figure 5.9). Each subinterval has lengthDx¼1/4.
These subintervals determine four subregions:R1,R2,R3, andR4. With each subregion, we can associate acircumscribedrectangle (Figure 5.10); that is, a rectangle whose base
2
x 1 0
y
f(x) = 2x
FIGURE 5.8
is the corresponding subinterval and whose height is the maximum value of f on that subinterval.
In this case, sincefisan increasing function, the maximum value offon each subinterval will occur whenxis the right-hand end point of the subinterval.
The areas of thecircumscribedrectangles (Figure 5.10) associated with regionsR1,R2,R3, andR4are 1/4f(1/4), 1/4f(2/4), 1/4f(3/4), and 1/4f(4/4), respectively. The area of each rectangle isan approximationto the area of its correspondingsubregion.
Thus, the sum of the areas of the circumscribed rectangles, denoted byF4(upper sum), is an approximation to the areaAof the triangle.
F4¼1 4f 1
4 þ1 4f 2
4 þ1 4f 3
4 þ1 4f 4
4
¼1 4 2 1
4 þ2 2 4 þ2 3
4 þ2 4 4
; ð ) fðxÞ ¼2xÞ
¼1 4
1 2þ2
2þ3 2þ4
2
¼10 8 ¼5
4 Using sigma notation, we can write
F4¼X4
i¼1
fðxiÞDx
Obviously,F4isgreater than the actual areaof the triangle, sinceit includes areas of shaded regionsthat are not in the triangle (Figure 5.10).
y
f(x) = 2x
R4 R3
R2
R1 1 4 0
X0 X1 X2 X3 X4 x
2
2
4 3
4 4
4 FIGURE 5.9
THE DEFINITE INTEGRAL AS AN AREA 145
Similarly, the areas of the four inscribed rectangles (Figure 5.11) associated withR1,R2,R3, and R4are14fð0Þ;14f 14 ;14f 24 ;and14f 34 , respectively. Their sum, denoted byF4 (lower sum), is also an approximation to the areaAof the triangle.(6)
F4¼1 4fð0Þ þ1
4f 1 4 þ1
4f 2 4 þ1
4f 3 4
¼1
4 2ð0Þ þ2 1 4 þ2 2
4 þ2 3 4
; ð ) fðxÞ ¼2xÞ
¼1 4 0þ1
2þ2 2þ3
2
¼6 8¼3
4 Using sigma notation, we can write
F4¼X4
i¼1
fðxi1ÞDx y
x f(x) = 2x
f(1) 4 f(2)
4 f(34) f(4) 4
41 2
4 3
4 4
0 4
FIGURE 5.10
(6)Sincefis an increasing function, the minimum value offon each subinterval will occur whenxis the left-hand end point.
In general, maximum or minimum values of a function, on each subinterval, may occur at any point in the subinterval.
Clearly,F4is less than the area of the triangle because the rectangles do not account for that portion of the triangle, which is not shaded. Note that
3
4¼F4AF4¼5 4
We say thatF4 is an approximation toA from below andF4 is an approximation toA from above.
If [0,1] is divided into more subintervals, better approximations toAwill occur. For example, let us usesix subintervals of equal lengthDx¼1=6. Then, the total area ofsix circumscribed rectangles(i.e.,the upper sum) is given by
F6¼1 6f 1
6 þ1 6f 2
6 þ1 6f 3
6 þ1 6f 4
6 þ1 6f 5
6 þ1 6f 6
6
¼1 6 2 1
6 þ2 2 6 þ2 3
6 þ2 4 6 þ2 5
6 þ2 6 6
¼1 6
1 3þ2
3þ3 3þ4
3þ5 3þ6
3
¼21 18¼7
6 y
f(x) = 2x
x f(3)
4
f(2) 4
f(1) 4 f(4)
4
0 1
4 2
4 3
4 4
4 f(0)
FIGURE 5.11
THE DEFINITE INTEGRAL AS AN AREA 147
and, the total area ofsix inscribed rectangles(i.e.,the lower sum) is given by
F¼1 6fð0Þ þ1
6f 1 6 þ1
6f 2 6 þ1
6f 3 6 þ1
6f 4 6 þ1
6f 5 6
¼1
6 2ð0Þ þ2 1 6 þ2 2
6 þ2 3 6 þ2 4
6 þ2 5 6
¼1 6 0þ1
3þ2 3þ3
3þ4 3þ5
3
¼15 18¼5
6
Note thatF6AF6and,with appropriate labeling, bothF6andF6will be of the form PfðxÞDx.(7)
More generally, if we divide [0,1] inton subintervals of equal lengthDx, thenDx¼1/nand the end points of the subintervals arex¼0, 1/n, 2/n,. . ., (n1)/nandn/n¼1 (see Figure 5.12).
y
f(x) = 2x
0 1n f( )
2n f( )
nn f( )
1n 2 n
nn
x
n–1 n FIGURE 5.12
(7)F6¼P6
i¼1fðxiÞDx;F6¼P5
i¼0fðxiÞDx.
The total area ofn circumscribed rectanglesis Fn¼1
nf 1 n þ1
nf 2 n þ1
nf 3
n þ. . .þ1
nf n n
¼1 n 2 1
n þ2 2 n þ2 3
n þ. . .þ2 n
n
; ½ ) fðxÞ ¼2x
¼ 2
n2½1þ2þ3þ. . .þn; ðby taking out 2=nfrom each termÞ
¼ 2 n2
nðnþ1Þ
2 ; since 1þ2þ. . .þn¼nðnþ1Þ 2
¼nþ1 n
ð1Þ
And forn inscribed rectangles, the total area determined by the subintervals is (see Figure 5.13) Fn¼1
nfð0Þ þ1 nf 1
n þ. . .þ1
nf 2 n . . .þ1
nf n1 n
¼1
n 2ð0Þ þ2 1 n þ2 2
n þ. . .þ2 n1
n
¼ 2
n2½0þ1þ2þ3þ. . .þ ðn1Þ
¼ 2 n2
ðn1Þn
2 ¼n1
n ; since 1þ2þ. . .þ ðn1Þ ¼ðn1ÞðnÞ 2
ð2Þ
From equations (1) and (2), we observe that bothFnandFnare sums of the formP fðxÞDx.
From the nature ofFnandFn, it is reasonable and indeed true to writeFnAFn. Asn becomes larger,FnandFnbecome better approximations toA from belowandfrom above, respectively. If we take the limit ofFnandFnasn!,1through positive integral values, we get
n! 1lim Fn¼ lim
n! 1
n1 n ¼ lim
n! 1 11
n
¼1; and
n! 1lim Fn¼ lim
n! 1
nþ1 n ¼ lim
n! 1 1þ1 n
¼1 SinceFnandFnboth have thesame common limit, we write
n! 1lim Fn¼ lim
n! 1Fn¼1 ð3Þ
and sinceFnAFn,we take this common limit to be the area of the triangle. Thus we get, the areaA¼1 square unit.This also agrees with our earlier finding.
Mathematically,the sums FnandFn,as well as their common limit have a meaning, which is independent of area. For the functionf(x)¼2x, over the interval [0, 1],we define the common limit of FnandFnto be the definite integral of f(x)¼2x, from x¼0to x¼1. Symbolically we write this as
ð1 0
fðxÞdx¼ ð1
0
2xdx¼1 ð4Þ
THE DEFINITE INTEGRAL AS AN AREA 149
The numbers 0 and 1 appearing with the integral signÐ
in equation (4) are calledthe limits of integration; 0 is thelower limitand 1 is theupper limit.
Two points must be mentioned about thedefinite integral:
(i) Aside from any geometrical interpretation (such as area)it is nothing more than a real number.
(ii) The definite integral is the limit of a sum of the formP fðxÞDx.
The definite integral of a function f(x) over an interval from x¼a to x¼b, where ab, is the common limit of the upper sum (i.e., Fn) and the lower sum (i.e., Fn), if it exists, and is written asÐb
afðxÞdx. In terms of the limiting process, we haveP
fðxÞDx!Ðb
bfðxÞdðxÞ. We take this limiting value as thedefinition of the definite integral. In particular, thedefinite integral also stands for the area under a curve, as discussed above. From a subdivision of the interval [a,b]
into finite portions of the formDx,the process of passage to the limit (asDx!0) is suggested by the use of the letter d in place ofD.
Note:It will be wrong to think that dxis an infinitely small quantity or an infinitesimal (i.e., a variable whose limit is 0) or that the definite integralÐb
afðxÞdxis the sum of an infinite number of infinitely small quantities.This type of thinking is quite misleading and it is a sign of being in the state of confusion. Hence, care must be taken to protect and preserve what we have carried out with precision. We now formally define the area in terms of the definite integral.
y
f(x) = 2x
f( )1n
1n 2 n
nn f( )nn–1
n–1 n
x 0
FIGURE 5.13