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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

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 =

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)

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)

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)

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.

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.

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→

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 .

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.

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 .

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.

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.

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.

ExampleFind the area, A, under f (x) = x2 between x = 0 and x = 1, usingx∗i = right end point of [xi , xi+1].

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.