Chapter 5 – Integrals 5.1 Areas and Distances We start with a problem – how can we calculate the area “under” a given function – ie, the area between the function and the x-axis? If the curve happens to be something easy – like a horizontal line - this isn’t too hard, but if it’s a curve, the problem becomes more difficult. Let’s start by drawing a rectangle that’s f(a) tall by (b – a) wide as an estimate: Area = f(a) (b – a) How can we get a better estimate? Let’s try 2 rectangles: Area = f(a) ∆x 1 + f(c) ∆x 2 How could we get an even better estimate? Ft .÷ HEI ÷nk
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Chapter 5 – Integrals 5.1 Areas and Distances We start with a problem – how can we calculate the area “under” a given function – ie, the area between the function and the x-axis? If the curve happens to be something easy – like a horizontal line - this isn’t too hard, but if it’s a curve, the problem becomes more difficult.
Let’s start by drawing a rectangle that’s f(a) tall by (b – a) wide as an estimate:
Area = f(a) (b – a) How can we get a better estimate? Let’s try 2 rectangles:
Area = f(a) ∆x1 + f(c) ∆x2 How could we get an even better estimate?
Ft.÷HEI÷nk
Ex: Estimate the area under the curve f(x) = x2 between x = 0 and x = 1 by constructing 8 rectangles (the book calls them “strips”) of equal width. This seems like a relatively simple situation, until you realize that you get different answers depending on whether you draw the rectangles from the left or from the right:
Using left endpoints, we get: f(0) !
" + f !"
!" + f #
"!" + f $
"!" + f %
"!" + f &
"!" + f '
"!" + f (
"!"
We can see this will be a lower bound, because all the rectangles fit completely under the curve. Using right endpoints, we get: f !"
!" + f #
"!" + f $
"!" + f %
"!" + f &
"!" + f '
"!" + f (
"!" + f 1 !
" We can see this will be an upper bound, because all the rectangles extend to above the curve. By computing both, we get: 0.2734375 < A < 0.3984375 Hmmm…if only we had a technique for looking at a progression of smaller and smaller rectangles….
ft:-
1
Definition: The Area A of the region S that lies under the graph of the continuous function f is the limit of the sum of the areas of approximating rectangles (using right endpoints): A = lim
-→/0- = lim-→/ 3 4! ∆4 + 3 4# ∆4 + ⋯+ 3(4-)∆4
1) It can be proven that this limit always exists. 2) It can also be shown that you get the same limit value A from using left endpoints. 3) In fact, you can choose any x-value in each interval to calculate the height. The values that are chosen - x1
*, x2*, …, xn
* - are called the sample points. Using sigma notation we get: A = lim
-→/3(4:)∆4-
:;! The Distance Problem My odometer is broken, but I’d still like to calculate how far I’ve driven using my speedometer and a stopwatch (held by my passenger, for safety.) I know D = R*T and I have the following data:
Time (s) 0 5 10 15
Speed (ft/s) 0 20 25 10 How can I estimate the distance I’ve travelled?
n = #
rectangle
him I,fcxisn
fcx ,) dx + F Cxddxt . .
tfcxn ) IX
Using Left Endpoints: 0 * 5 + 20 * 5 + 25 * 5 = 225 ft Using Right Endpoints: 20 * 5 + 25 * 5 + 10 * 5 = 275 ft 1) How can we improve our estimate? 2) Do we know whether the true distance lies between the two estimates?
tsp^# seconds
More frequent speed checks
Ex odometer
No
5.2 The Definite Integral The limit of the sum from 5.1: lim-→/
' (! ∆( + ' (# ∆( + ⋯+ '((-)∆( = lim-→/
'((:)∆(-:;!
where ∆( = <=>- , can be written as:
' ( 7(<
>
and is called the definite integral of f from a to b, as long as the limit exits and is the same for all choices of xi. If the limit exists, then f is said to be integrable over [a,b]
Properties of the Integral 1) 87(<
A = c(b – a) where c is any constant 2) ' ( + : ( 7( =<
> ' ( 7(<> + : ( 7(<
> 3) 8' ( 7(<
> = c ' ( 7(<> where c is any constant
4) [' ( − : ( ]7(<
> = ' ( 7(<> - : ( 7(<
> 5) ' ( 7(F
> + ' ( 7(<F = ' ( 7(<
> 6) If f(x) > 0 for a < x < b, then ' ( 7(<
> > 0 7) If f(x) > g(x) for a < x < b, then ' ( 7(<
> > : ( 7(<>
8) If m < f(x) < M for a < x < b, then m (b – a) < ' ( 7(<
> ≤ @(A − B)
rwsum
Boata
Ex: (#33) The graph of f is shown. Evaluate each integral by interpreting it in terms of areas.
a) ' ( 7(#
K d) ' ( 7(L
K
i.ii AN'#(
+0
-
=i+i+i+÷+t5h ,¥'
,
= 4 T
10 + C- 8) = 2
5.3 The Fundamental Theorem of Calculus (Cue exciting music)
The Fundamental Theorem of Calculus, Part I If f is continuous on [a,b], then the function g defined by g(x) = ' E 7EN
> a < x < b is continuous on [a,b] and differentiable on (a,b), and g’(x) = f(x).
Ex: Find OON sec E7ENS
! (Hint: You’ll need the Chain Rule)
The Fundamental Theorem of Calculus, Part 2 If f is continuous on [a, b], then ' ( 7(<
> = F(b) – F(a) where F is any antiderivative of f. That is, a function s. t. F’ = f.
Why can it be *any* antiderivative of f?
= seek4) . 4×3
The C 'scancel out.
So, differentiation and integration are inverse processes. Ex: Evaluate these integrals 1) 1 + ( + 3(# 7(!
K 2) 4 − E E7E%
K Note: Be careful when rushing in to evaluate integrals. Look at this example:
1(# 7(
$
=!
It seems like it would be easy to find an antiderivative and evaluate it, but first – is this function continuous over [-1, 3]?
= x + I +3¥ ] to
F ( i ) - FO :
= f + ±¥¥j - [ o + It sgj= [ i + E + I] - [D= 2.5
= 64 (4 re . tr ) dt
±I±÷IItI¥± IIIIII:*:*'s¥;**¥= 6¥ - 6¥ = ¥8
⇒ n*±÷¥n⇒K¥¥¥e¥s±÷
No,
not at 0.
5.4 Indefinite Integrals Definite vs. Indefinite Integrals: 2"$"$
J is a definite integral. It’s a string of symbols that represent a number: 2"$"$
J = x2 ]03 = 32 – 02 = 9 2"$"is an indefinite integral. It’s a string of symbols that represent a