Download Math 71 – 1.1

Survey
yes no Was this document useful for you?
   Thank you for your participation!

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Document related concepts

John Wallis wikipedia, lookup

Pi wikipedia, lookup

Function of several real variables wikipedia, lookup

Infinitesimal wikipedia, lookup

Series (mathematics) wikipedia, lookup

Path integral formulation wikipedia, lookup

Divergent series wikipedia, lookup

ItΓ΄ calculus wikipedia, lookup

Riemann integral wikipedia, lookup

Multiple integral wikipedia, lookup

Lebesgue integration wikipedia, lookup

History of calculus wikipedia, lookup

Integral wikipedia, lookup

Fundamental theorem of calculus wikipedia, lookup

Transcript
Math 140
5.3 – The Definite Integral and the
Fundamental Theorem of Calculus
1
Q: How can we find the area under any curve?
For example, let’s try to find the area under
𝑓 π‘₯ = π‘₯ 3 βˆ’ 9π‘₯ 2 + 25π‘₯ βˆ’ 19
2
What is
the area
of this?
3
1 rectangle
4
2 rectangles
5
4 rectangles
6
50 rectangles
7
If the left sides of our rectangles are at π‘₯1 , π‘₯2 , …, π‘₯𝑛
and each rectangle has width Ξ”π‘₯,
then the areas of the rectangles are
𝑓 π‘₯1 Ξ”π‘₯, 𝑓 π‘₯2 Ξ”π‘₯, …, 𝑓 π‘₯𝑛 Ξ”π‘₯.
So, the sum of the areas of the rectangles (called a
Riemann sum) is:
𝑓 π‘₯1 Ξ”π‘₯ + 𝑓 π‘₯2 Ξ”π‘₯ + β‹― + 𝑓 π‘₯𝑛 Ξ”π‘₯
= 𝑓 π‘₯1 + 𝑓 π‘₯2 + β‹― + 𝑓 π‘₯𝑛 Ξ”π‘₯
8
So, the sum of the areas of the rectangles (called a
Riemann sum) is:
𝑓 π‘₯1 Ξ”π‘₯ + 𝑓 π‘₯2 Ξ”π‘₯ + β‹― + 𝑓 π‘₯𝑛 Ξ”π‘₯
= 𝑓 π‘₯1 + 𝑓 π‘₯2 + β‹― + 𝑓 π‘₯𝑛 Ξ”π‘₯
9
So, the sum of the areas of the rectangles (called a
Riemann sum) is:
𝑓 π‘₯1 Ξ”π‘₯ + 𝑓 π‘₯2 Ξ”π‘₯ + β‹― + 𝑓 π‘₯𝑛 Ξ”π‘₯
= 𝑓 π‘₯1 + 𝑓 π‘₯2 + β‹― + 𝑓 π‘₯𝑛 Ξ”π‘₯
10
As the number of rectangles (𝑛) increases, the
above sum gets closer to the true area.
That is,
Area under curve =
π’π’Šπ’Ž 𝒇 π’™πŸ + 𝒇 π’™πŸ + β‹― + 𝒇 𝒙𝒏
𝐧→+∞
πœŸπ’™
11
𝑏
𝑓 π‘₯ 𝑑π‘₯ = lim 𝑓 π‘₯1 + 𝑓 π‘₯2 + β‹― + 𝑓 π‘₯𝑛
π‘Ž
nβ†’+∞
Ξ”π‘₯
The above integral is called a definite integral, and π‘Ž
and 𝑏 are called the lower and upper limits of
integration.
12
The Fundamental Theorem of Calculus
If 𝑓(π‘₯) is continuous on the interval π‘Ž ≀ π‘₯ ≀ 𝑏,
then
𝒃
𝒇 𝒙 𝒅𝒙 = 𝑭 𝒃 βˆ’ 𝑭(𝒂)
𝒂
where 𝐹(π‘₯) is any antiderivative of 𝑓(π‘₯) on π‘Ž ≀
π‘₯ ≀ 𝑏.
13
Ex 1.
Evaluate the given definite integrals using the
fundamental theorem of calculus.
1
2π‘₯ 2 𝑑π‘₯
βˆ’1
14
Ex 1.
Evaluate the given definite integrals using the
fundamental theorem of calculus.
1
2π‘₯ 2 𝑑π‘₯
βˆ’1
15
Ex 1 (cont).
Evaluate the given definite integrals using the
fundamental theorem of calculus.
1
(𝑒 2π‘₯ βˆ’ π‘₯) 𝑑π‘₯
0
16
Ex 1 (cont).
Evaluate the given definite integrals using the
fundamental theorem of calculus.
1
(𝑒 2π‘₯ βˆ’ π‘₯) 𝑑π‘₯
0
17
Note: Definite integrals can evaluate to
negative numbers. For example,
2
1
(1
βˆ’
π‘₯)
𝑑π‘₯
=
βˆ’
1
2
18
Just like with indefinite integrals, terms can be integrated
separately, and constants can be pulled out.
𝑏
𝑏
𝑓 π‘₯ ± 𝑔(π‘₯) 𝑑π‘₯ =
π‘Ž
𝑏
𝑏
π‘˜π‘“ π‘₯ 𝑑π‘₯ = π‘˜
π‘Ž
𝑏
𝑓 π‘₯ 𝑑π‘₯ ±
π‘Ž
𝑔(π‘₯) 𝑑π‘₯
π‘Ž
𝑓 π‘₯ 𝑑π‘₯
π‘Ž
Also,
π‘Ž
𝑓 π‘₯ 𝑑π‘₯ = 0
π‘Ž
π‘Ž
𝑏
𝑓 π‘₯ 𝑑π‘₯ = βˆ’
𝑏
𝑓 π‘₯ 𝑑π‘₯
π‘Ž
19
𝑏
𝑐
𝑓 π‘₯ 𝑑π‘₯ =
π‘Ž
𝑏
𝑓 π‘₯ 𝑑π‘₯ +
π‘Ž
𝑓 π‘₯ 𝑑π‘₯
𝑐
(here, 𝑐 is any real number)
20
𝑏
𝑐
𝑓 π‘₯ 𝑑π‘₯ =
π‘Ž
𝑏
𝑓 π‘₯ 𝑑π‘₯ +
π‘Ž
𝑓 π‘₯ 𝑑π‘₯
𝑐
(here, 𝑐 is any real number)
=
21
𝑏
𝑐
𝑓 π‘₯ 𝑑π‘₯ =
π‘Ž
𝑏
𝑓 π‘₯ 𝑑π‘₯ +
π‘Ž
𝑓 π‘₯ 𝑑π‘₯
𝑐
(here, 𝑐 is any real number)
=
+
22
Ex 2.
Let 𝑓(π‘₯) and 𝑔(π‘₯) be functions that are continuous on the interval
βˆ’ 3 ≀ π‘₯ ≀ 5 and that satisfy
5
5
𝑓 π‘₯ 𝑑π‘₯ = 2
5
𝑔 π‘₯ 𝑑π‘₯ = 7
βˆ’3
βˆ’3
𝑓 π‘₯ 𝑑π‘₯ = βˆ’8
1
Use this information along with the rules definite integrals to evaluate
the following.
5
2𝑓 π‘₯ βˆ’ 3𝑔 π‘₯
𝑑π‘₯
βˆ’3
1
𝑓 π‘₯ 𝑑π‘₯
βˆ’3
23
Ex 3.
Evaluate:
1
8π‘₯ π‘₯ 2 + 1
3
𝑑π‘₯
0
24
Ex 3.
Evaluate:
1
8π‘₯ π‘₯ 2 + 1
3
𝑑π‘₯
0
25
Net Change
If we have 𝑄′(π‘₯), then we can calculate the net
change in 𝑄(π‘₯) as π‘₯ goes from π‘Ž to 𝑏 with an
integral:
𝑏
𝑄 β€² π‘₯ 𝑑π‘₯
𝑄 𝑏 βˆ’π‘„ π‘Ž =
π‘Ž
26
Net Change
Ex 4.
Suppose marginal cost is 𝐢 β€² π‘₯ = 3 π‘₯ βˆ’ 4 2
dollars per unit when the level of production is π‘₯
units. By how much will the total manufacturing
cost increase if the level of production is raised
from 6 units to 10 units?
27