ch 5.1: area - kennesaw state universityfacultyweb.kennesaw.edu/ykang4/file_1/math2202/... · ch...
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Ch 5.1: Area
In this section, we will
I define finite summation
I estimate the area of a positive function on a finite interval
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Finite Sums: Sigma Notation∑
n∑k=1
ak = a1 + a2 + a3 + · · ·+ an−1 + an.
∑(reads capital sigma) stands for sum. The index of
summation k tells you where the sum begins and where it ends.Examples)
1.∑5
k=1 k = 1 + 2 + 3 + 4 + 5.
2.∑6
k=3 2k =.
3.∑3
k=1(−1)kk2 =
4.∑5
k=3k2
k+1 =
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Rules for finite Sums
Theorem
1.∑n
k=1(ak + bk) =∑n
k=1 ak +∑n
k=1 bk
2.∑n
k=1(ak − bk) =∑n
k=1 ak −∑n
k=1 bk
3.∑n
k=1 c · ak = c ·∑n
k=1 ak for c constant.
4.∑n
k=1 c = n · c
Example) Evaluate∑3
k=1(3k − k2 + 2)
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Useful sums to know/remember:
Theorem
I∑n
k=1 k = n(n+1)2
I∑n
k=1 k2 = n(n+1)(2n+1)6
I∑n
k=1 k3 =(n(n+1)
2
)2Examples)
1.∑n
k=1(3k − k2)
2.∑7
k=1 k(2k + 1)
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More Examples
Theorem
I∑n
k=1 k = n(n+1)2
I∑n
k=1 k2 = n(n+1)(2n+1)6
I∑n
k=1 k3 =(n(n+1)
2
)21. (1 + 2 + 3 + · · ·+ 15)
2. (36 + 49 + 64 + · · · n2)
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The area problemFirst, consider the positive function f (x) = x2 from [0, 1].
We shall estimate the finite area under f (x) on [0, 1] (let’s call itS) by estimating with rectangles that contain the actual areaunder f , the shaded region.
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I With 1 rectangle containing the actual area under f :
I With 2 rectangles:
I With 3 rectangles:
I With n rectangles.
Estimating the area under f this way always produces the sumlarger/smaller than the actual area. Thus, the sum is called theupper sum, denoted by Sn.
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When increasing the number of the rectangles
Remarks)
1. When we increase the number of the rectangles, n, toestimate the actual area, the estimated area, Sn, becomes abetter/worse approximation the actual area under thefunction.
2. Then to achieve the actual area under f , we need to letn→
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The area UNDER the curveConsider the same function f (x) = x2 from [0, 1].
Another way of estimating the area under f (x) on [0, 1] is byestimating with rectangles that are contained in the actual areaunder f , the shaded region S .
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I With 1 rectangle that is contained in the region S :
I With 2 rectangles:
I With 3 rectangles:
I With n rectangles.
Estimating the area under f this way always produces the sumlarger/smaller than the actual area. Thus, the sum is called thelower sum, denoted by sn.
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When increasing the number of the rectangles
Remarks)
1. When we increase the number of the rectangles (n) in orderto estimate the actual area, the estimated area, sn, becomes abetter/worse approximation the actual area under thefunction.
2. Then to achieve the actual area under f , we need to letn→
3. Thus, we can choose either endpoints (left or right) to findthe actual area S .
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ExampleConsider f (x) = x2 on [0, 1]. Approximate the area under thecurve f on [0, 1] by setting up (do not solve) the Upper sum andthe Lower sum. Let the number of rectangles n = 4.
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ExampleLet f (x) = 1
x on [1, 5]. Approximate the area under the curve f on[1, 5] by setting up (do not solve) the Upper sum as well as theLower sum. Let the number of rectangles n = 4.
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The area A of a region
TheoremThe area A under the graph of a positive continuous function f isthe limit of the sum of the areas of approximating rectangles:
A = limn→∞
[f (c1)∆x + f (c2)∆x + · · ·+ f (cn)∆x ]
= limn→∞
[f (C1)∆x + f (C2)∆x + · · ·+ f (Cn)∆x ]
= limn→∞
[f (x∗1 )∆x + f (x∗2 )∆x + · · ·+ f (x∗n )∆x ]
where ci is a value in [xi−1, xi ] such that f (ci ) gives the absoluteminimum value of f in this interval, Ci is a value in [xi−1, xi ] suchthat f (Ci ) gives the absolute maxmum value of f in this interval,and x∗i is any number in the i-th sub-interval [xi−1, xi ]. We call x∗ithe sample points.
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ExampleFind the area, A, under f (x) = x2 between x = 0 and x = 1, usingx∗i = right end point of [xi , xi+1].
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Class ExerciseEstimate the area A, under f (x) = x2 between x = 0 and x = 2for n = 4, by using lower and upper sum. That is, find sn and Sn
for n = 4.