Professor Switkes

Riemann Sums, Definite Integrals, Fundamental Theorem of Calculus

Slide Duration:

Table of Contents

Section 1: Overview of Functions

Review of Functions

26m 29s

Intro

0:00

What is a Function

0:10

Domain and Range

0:21

Vertical Line Test

0:31

Example: Vertical Line Test

0:47

Function Examples

1:57

Example: Squared

2:10

Example: Natural Log

2:41

Example: Exponential

3:21

Example: Not Function

3:54

Odd and Even Functions

4:39

Example: Even Function

5:10

Example: Odd Function

5:53

Odd and Even Examples

6:48

Odd Function

6:55

Even Function

8:43

Increasing and Decreasing Functions

10:15

Example: Increasing

10:42

Example: Decreasing

10:55

Increasing and Decreasing Examples

11:41

Example: Increasing

11:48

Example: Decreasing

12:33

Types of Functions

13:32

Polynomials

13:45

Powers

14:06

Trigonometric

14:34

Rational

14:50

Exponential

15:13

Logarithmic

15:29

Lecture Example 1

15:55

Lecture Example 2

17:51

Additional Example 3

-1

Additional Example 4

-2

Compositions of Functions

12m 29s

Intro

0:00

Compositions

0:09

Alternative Notation

0:32

Three Functions

0:47

Lecture Example 1

1:19

Lecture Example 2

3:25

Lecture Example 3

6:45

Additional Example 4

-1

Additional Example 5

-2

Section 2: Limits

Average and Instantaneous Rates of Change

20m 59s

Intro

0:00

Rates of Change

0:11

Average Rate of Change

0:21

Instantaneous Rate of Change

0:33

Slope of the Secant Line

0:46

Slope of the Tangent Line

1:00

Lecture Example 1

1:14

Lecture Example 2

6:36

Lecture Example 3

11:30

Additional Example 4

-1

Additional Example 5

-2

Limit Investigations

22m 37s

Intro

0:00

What is a Limit?

0:10

Lecture Example 1

0:56

Lecture Example 2

5:28

Lecture Example 3

9:27

Additional Example 4

-1

Additional Example 5

-2

Algebraic Evaluation of Limits

28m 19s

Intro

0:00

Evaluating Limits

0:09

Lecture Example 1

1:06

Lecture Example 2

5:16

Lecture Example 3

8:15

Lecture Example 4

12:58

Additional Example 5

-1

Additional Example 6

-2

Formal Definition of a Limit

23m 39s

Intro

0:00

Formal Definition

0:13

Template

0:55

Epsilon and Delta

1:24

Lecture Example 1

1:40

Lecture Example 2

9:20

Additional Example 3

-1

Additional Example 4

-2

Continuity and the Intermediate Value Theorem

19m 9s

Intro

0:00

Continuity

0:13

Continuous

0:16

Discontinuous

0:37

Intermediate Value Theorem

0:52

Example

1:22

Lecture Example 1

2:58

Lecture Example 2

9:02

Additional Example 3

-1

Additional Example 4

-2

Section 3: Derivatives, part 1

Limit Definition of the Derivative

22m 52s

Intro

0:00

Limit Definition of the Derivative

0:11

Three Versions

0:13

Lecture Example 1

1:02

Lecture Example 2

4:33

Lecture Example 3

6:49

Lecture Example 4

10:11

Additional Example 5

-1

Additional Example 6

-2

The Power Rule

26m 1s

Intro

0:00

Power Rule of Differentiation

0:14

Power Rule with Constant

0:41

Sum/Difference

1:15

Lecture Example 1

1:59

Lecture Example 2

6:48

Lecture Example 3

11:22

Additional Example 4

-1

Additional Example 5

-2

The Product Rule

14m 54s

Statement of the Product Rule

0:08

Lecture Example 1

0:41

Lecture Example 2

2:27

Lecture Example 3

5:03

Additional Example 4

-1

Additional Example 5

-2

The Quotient Rule

19m 17s

Intro

0:00

Statement of the Quotient Rule

0:07

Carrying out the Differentiation

0:23

Quotient Rule in Words

1:00

Lecture Example 1

1:19

Lecture Example 2

4:23

Lecture Example 3

8:00

Additional Example 4

-1

Additional Example 5

-2

Applications of Rates of Change

17m 43s

Intro

0:00

Rates of Change

0:11

Lecture Example 1

0:44

Lecture Example 2

5:16

Lecture Example 3

7:38

Additional Example 4

-1

Additional Example 5

-2

Trigonometric Derivatives

26m 58s

Intro

0:00

Six Basic Trigonometric Functions

0:11

Patterns

0:47

Lecture Example 1

1:18

Lecture Example 2

7:38

Lecture Example 3

12:15

Lecture Example 4

14:25

Additional Example 5

-1

Additional Example 6

-2

The Chain Rule

23m 47s

Intro

0:00

Statement of the Chain Rule

0:09

Chain Rule for Three Functions

0:27

Lecture Example 1

1:00

Lecture Example 2

4:34

Lecture Example 3

7:23

Additional Example 4

-1

Additional Example 5

-2

Inverse Trigonometric Functions

27m 5s

Intro

0:00

Six Basic Inverse Trigonometric Functions

0:10

Lecture Example 1

1:11

Lecture Example 2

8:53

Lecture Example 3

12:37

Additional Example 4

-1

Additional Example 5

-2

Equation of a Tangent Line

15m 52s

Intro

0:00

Point Slope Form

0:10

Lecture Example 1

0:47

Lecture Example 2

3:15

Lecture Example 3

6:10

Additional Example 4

-1

Additional Example 5

-2

Section 4: Derivatives, part 2

Implicit Differentiation

30m 5s

Intro

0:00

Purpose

0:09

Implicit Function

0:20

Lecture Example 1

0:32

Lecture Example 2

7:14

Lecture Example 3

11:22

Lecture Example 4

16:43

Additional Example 5

-1

Additional Example 6

-2

Higher Derivatives

13m 16s

Intro

0:00

Notation

0:08

First Type

0:19

Second Type

0:54

Lecture Example 1

1:41

Lecture Example 2

3:15

Lecture Example 3

4:57

Additional Example 4

-1

Additional Example 5

-2

Logarithmic and Exponential Function Derivatives

17m 42s

Intro

0:00

Essential Equations

0:12

Lecture Example 1

1:34

Lecture Example 2

2:48

Lecture Example 3

5:54

Additional Example 4

-1

Additional Example 5

-2

Hyperbolic Trigonometric Function Derivatives

14m 30s

Intro

0:00

Essential Equations

0:15

Six Basic Hyperbolic Trigc Functions

0:32

Six Basic Inverse Hyperbolic Trig Functions

1:21

Lecture Example 1

1:48

Lecture Example 2

3:45

Lecture Example 3

7:09

Additional Example 4

-1

Additional Example 5

-2

Related Rates

29m 5s

Intro

0:00

What Are Related Rates?

0:08

Lecture Example 1

0:35

Lecture Example 2

5:25

Lecture Example 3

11:54

Additional Example 4

-1

Additional Example 5

-2

Linear Approximation

23m 52s

Intro

0:00

Essential Equations

0:09

Linear Approximation (Tangent Line)

0:18

Example: Graph

1:18

Differential (df)

2:06

Delta F

5:10

Lecture Example 1

6:38

Lecture Example 2

11:53

Lecture Example 3

15:54

Additional Example 4

-1

Additional Example 5

-2

Section 5: Application of Derivatives

Absolute Minima and Maxima

18m 57s

Intro

0:00

Minimums and Maximums

0:09

Absolute Minima and Maxima (Extrema)

0:53

Critical Points

1:25

Lecture Example 1

2:58

Lecture Example 2

6:57

Lecture Example 3

10:02

Additional Example 4

-1

Additional Example 5

-2

Mean Value Theorem and Rolle's Theorem

20m

Intro

0:00

Theorems

0:09

Mean Value Theorem

0:13

Graphical Explanation

0:36

Rolle's Theorem

2:06

Graphical Explanation

2:28

Lecture Example 1

3:36

Lecture Example 2

6:33

Lecture Example 3

9:32

Additional Example 4

-1

Additional Example 5

-2

First Derivative Test, Second Derivative Test

27m 11s

Intro

0:00

Local Minimum and Local Maximum

0:14

Example

1:01

First and Second Derivative Test

1:26

First Derivative Test

1:36

Example

2:00

Second Derivative Test (Concavity)

2:58

Example: Concave Down

3:15

Example: Concave Up

3:54

Inconclusive

4:19

Lecture Example 1

5:23

Lecture Example 2

12:03

Lecture Example 3

15:54

Additional Example 4

-1

Additional Example 5

-2

L'Hopital's Rule

23m 9s

Intro

0:00

Using L'Hopital's Rule

0:09

Informal Definition

0:34

Lecture Example 1

1:27

Lecture Example 2

4:00

Lecture Example 3

5:40

Lecture Example 4

9:38

Additional Example 5

-1

Additional Example 6

-2

Curve Sketching

40m 16s

Intro

0:00

Collecting Information

0:15

Domain and Range

0:17

Intercepts

0:21

Symmetry Properties (Even/Odd/Periodic)

0:33

Asymptotes (Vertical/Horizontal/Slant)

0:45

Critical Points

1:15

Increasing/Decreasing Intervals

1:24

Inflection Points

1:38

Concave Up/Down

1:52

Maxima/Minima

2:03

Lecture Example 1

2:58

Lecture Example 2

10:52

Lecture Example 3

17:55

Additional Example 4

-1

Additional Example 5

-2

Applied Optimization

25m 37s

Intro

0:00

Real World Problems

0:08

Sketch

0:11

Interval

0:20

Rewrite in One Variable

0:26

Maximum or Minimum

0:34

Critical Points

0:42

Optimal Result

0:52

Lecture Example 1

1:05

Lecture Example 2

6:12

Lecture Example 3

13:31

Additional Example 4

-1

Additional Example 5

-2

Newton's Method

25m 13s

Intro

0:00

Approximating Using Newton's Method

0:10

Good Guesses for Convergence

0:32

Lecture Example 1

0:49

Lecture Example 2

4:21

Lecture Example 3

7:59

Additional Example 4

-1

Additional Example 5

-2

Section 6: Integrals

Approximating Areas and Distances

36m 50s

Intro

0:00

Three Approximations

0:12

Right Endpoint, Left Endpoint, Midpoint

0:22

Formulas

1:05

Velocity and Distance

1:35

Lecture Example 1

2:28

Lecture Example 2

12:10

Lecture Example 3

19:43

Additional Example 4

-1

Additional Example 5

-2

Riemann Sums, Definite Integrals, Fundamental Theorem of Calculus

22m 2s

Intro

0:00

Important Equations

0:22

Riemann Sum

0:28

Integral

1:58

Integrand

2:35

Limits of Integration (Upper Limit, Lower Limit)

2:43

Other Equations

3:05

Fundamental Theorem of Calculus

4:00

Lecture Example 1

5:04

Lecture Example 2

10:43

Lecture Example 3

13:52

Additional Example 4

-1

Additional Example 5

-2

Substitution Method for Integration

23m 19s

Intro

0:00

U-Substitution

0:13

Important Equations

0:30

Purpose

0:36

Lecture Example 1

1:30

Lecture Example 2

6:17

Lecture Example 3

9:00

Lecture Example 4

11:24

Additional Example 5

-1

Additional Example 6

-2

Section 7: Application of Integrals, part 1

Area Between Curves

19m 59s

Intro

0:00

Area Between Two Curves

0:12

Graphic Description

0:34

Lecture Example 1

1:44

Lecture Example 2

5:39

Lecture Example 3

8:45

Additional Example 4

-1

Additional Example 5

-2

Volume by Method of Disks and Washers

24m 22s

Intro

0:00

Important Equations

0:16

Equation 1: Rotation about x-axis (disks)

0:27

Equation 2: Two curves about x-axis (washers)

3:38

Equation 3: Rotation about y-axis

5:31

Lecture Example 1

6:05

Lecture Example 2

8:28

Lecture Example 3

11:55

Additional Example 4

-1

Additional Example 5

-2

Volume by Method of Cylindrical Shells

30m 29s

Intro

0:00

Important Equations

0:50

Equation 1: Rotation about y-axis

1:04

Equation 2: Rotation about y-axis (2 curves)

7:34

Equation 3: Rotation about x-axis

8:15

Lecture Example 1

8:57

Lecture Example 2

14:26

Lecture Example 3

18:15

Additional Example 4

-1

Additional Example 5

-2

Average Value of a Function

16m 31s

Intro

0:00

Important Equations

0:11

Origin of Formula

0:34

Lecture Example 1

2:51

Lecture Example 2

5:30

Lecture Example 3

8:13

Additional Example 4

-1

Additional Example 5

-2

Section 8: Extra

Graphs of f, f', f''

23m 58s

Intro

0:00

Slope Function of f(x)

0:41

Slope is Zero

0:53

Slope is Positive

1:03

Slope is Negative

1:13

Slope Function of f'(x)

1:31

Slope is Zero

1:42

Slope is Positive

1:48

Slope is Negative

1:54

Lecture Example 1

2:23

Lecture Example 2

8:06

Lecture Example 3

12:36

Additional Example 4

-1

Additional Example 5

-2

Slope Fields for Differential Equations

18m 32s

Intro

0:00

Things to Remember

0:13

Graphic Description

0:42

Lecture Example 1

1:44

Lecture Example 2

6:59

Lecture Example 3

9:46

Additional Example 4

-1

Additional Example 5

-2

Separable Differential Equations

17m 4s

Intro

0:00

Differential Equations

0:10

Focus on Exponential Growth/Decay

0:27

Separating Variables

0:47

Lecture Example 1

1:35

Lecture Example 2

6:41

Lecture Example 3

9:36

Additional Example 4

-1

Additional Example 5

-2

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For more information, please see full course syllabus of College Calculus: Level I

College Calculus: Level I Riemann Sums, Definite Integrals, Fundamental Theorem of Calculus

Section 6: Integrals: Lecture 2 | 22:02 min

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Lecture Comments (10)

	0 answers Post by Israel Haile on October 23, 2015 I loved your methodology ,I wanna encourage you, I am 99.99% satisfied with your class,Stay Blessed !!!
	0 answers Post by Valeriya Pinkhasova on February 5, 2015 I am unable to watch additional example. Everytime I press on it ,lecture starts from the beginning .
	0 answers Post by Jason Kim on November 5, 2014 wait for the antideriviative of the lecture example 2 how do you exactly know that anti deriviative of 2x-5 is just x^2-5X. It could also be x^2-5x plus some number because the number just disappears when we derive it. So it could be x^2-5x+9 and the deriviative would be also 2x-5. How can you assume that the antideriviative is just x^2-5x?
	0 answers Post by Johnny Zamora on January 14, 2014 She is Amazing
	0 answers Post by Jose Gonzalez-Gigato on November 10, 2013 'Additional Example 5' was excellent!
	1 answer Last reply by: Martina Alvarez Fri Dec 9, 2011 4:28 AM Post by Martina Alvarez on December 9, 2011 Ex: 2 13 min. =(36-30) - (1-5) =6 - (-4) =10 not 8
	0 answers Post by David Bascom on September 4, 2011 In lecture example 3 where did 3xsquared come from?
	1 answer Last reply by: Jose Gonzalez-Gigato Sun Nov 10, 2013 11:30 AM Post by Okwudili Ezeh on July 21, 2011 The graph was supposed to be a sine graph not a cosine graph.

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Riemann Sums, Definite Integrals, Fundamental Theorem of Calculus

Geometrically, Riemann sums represent sums of rectangle approximations.
The definite integral is a limit of Riemann sums.
For very simple functions, it is possible to directly compute Riemann sums and then take the limit.
Though this topic is very important theoretically, in practice we will compute integrals by using the First Fundamental Theorem of Calculus.

Riemann Sums, Definite Integrals, Fundamental Theorem of Calculus

∫₁⁵ dx

∫₁⁵ dx = x |₁⁵
∫₁⁵ dx = 5 − 1

∫₁⁵ dx = 4

∫₀^π cosx (sinx)³ dx

u = sinx
du = cosx dx
∫₀^π cosx (sinx)³ dx = ∫₀^π u³ du
∫₀^π cosx (sinx)³ dx = [(u⁴)/4] |₀^π
Remember, the limits of the integral are dependent on x. We must plug back in for x before using the original limits
∫₀^π cosx (sinx)³ dx = [((sinx)⁴)/4] |₀^π
∫₀^π cosx (sinx)³ dx = [((sinπ)⁴)/4] − [((sin0)⁴)/4]
∫₀^π cosx (sinx)³ dx = [0/4] − [0/4]

∫₀^π cosx (sinx)³ dx = 0

∫_−[(π)/2]^[(π)/2] 5 cosx dx

∫_−[(π)/2]^[(π)/2] 5 cosx dx = 5 sinx |_−[(π)/2]^[(π)/2]
∫_−[(π)/2]^[(π)/2] 5 cosx dx = 5 sin[(π)/2] − 5 sin[(−π)/2]
∫_−[(π)/2]^[(π)/2] 5 cosx dx = 5 − (−5)

∫_−[(π)/2]^[(π)/2] 5 cosx dx = 10

Confirm that ∫₋₂² x² dx = 2 ∫₀² x² dx

∫₋₂² x² dx = [(x³)/3] |₋₂²
∫₋₂² x² dx = [(2³)/3] − [((−2)³)/3]
∫₋₂² x² dx = [8/3] + [8/3]
∫₋₂² x² dx = [16/3]
2 ∫₀² x² dx = 2 [(x³)/3] |₀²
2 ∫₀² x² dx = 2 [(2³)/3] − 2[(0³)/3]
2 ∫₀² x² dx = [16/3]
This is a property of integrals involving even functions
∫_−a^a f(x) dx = 2 ∫₀^a f(x) dx if f(x) is even

Yes, ∫₋₂² x² dx = 2 ∫₀² x² dx

∫₀^ln5 e^2x dx

u = 2x
du = 2 dx
∫₀^ln5 e^5x dx = [1/2] ∫₀^ln5 e^u du
∫₀^ln5 e^5x dx = [1/2] e^u |₀^ln5
∫₀^ln5 e^5x dx = [1/2] e^2x |₀^ln5
∫₀^ln5 e^5x dx = [1/2] e^{2 ln5} − [1/2] e²⁽⁰⁾
∫₀^ln5 e^5x dx = [1/2] e^ln5² − [1/2] (1)
∫₀^ln5 e^5x dx = [25/2] − [1/2]

∫₀^ln5 e^5x dx = 12

∫₋₁¹ x² + 3x + 1 dx

∫₋₁¹ x² + 3x + 1 dx = ∫₋₁¹ x² dx + ∫₋₁¹ 3x dx + ∫₋₁¹ dx
∫₋₁¹ x² + 3x + 1 dx = [(x³)/3] |₋₁¹ + [(3x²)/2] |₋₁¹ + x |₋₁¹
∫₋₁¹ x² + 3x + 1 dx = [1/3] + [1/3] + [3/2] − [3/2] + 1 + 1

∫₋₁¹ x² + 3x + 1 dx = [8/3]

∫₋₁¹ x³ dx

∫₋₁¹ x³ dx = [(x⁴)/4] |₋₁¹
∫₋₁¹ x³ dx = [1/4] − [1/4]
Property of odd functions
∫_−a^a f(x) = 0 if f(x) is odd

∫₋₁¹ x³ dx = 0

∫₀² [1/(√{4 − x²})] dx

∫₀² [1/(√{4 − x²})] dx = ∫₀² [1/(√{2² − x²})] dx
∫₀² [1/(√{4 − x²})] dx = sin⁻¹ [x/2] |₀²
∫₀² [1/(√{4 − x²})] dx = sin⁻¹ 1 − sin⁻¹ 0
∫₀² [1/(√{4 − x²})] dx = [(π)/2] − 0

∫₀² [1/(√{4 − x²})] dx = [(π)/2]

∫₀¹ [1/(√{4 − x²})] dx + ∫₁² [1/(√{4 − x²})] dx

∫₀¹ [1/(√{4 − x²})] dx + ∫₁² [1/(√{4 − x²})] dx = sin⁻¹ [x/2] |₀¹ + sin⁻¹ [x/2] |₁²
∫₀¹ [1/(√{4 − x²})] dx + ∫₁² [1/(√{4 − x²})] dx = sin⁻¹ [1/2] − sin⁻¹ 0 + sin⁻¹ 1 − sin⁻¹ [1/2]
∫₀¹ [1/(√{4 − x²})] dx + ∫₁² [1/(√{4 − x²})] dx = sin⁻¹ 1 − sin⁻¹ 0
Another property of integrals.
∫_a^b f(x) dx + ∫_b^c f(x) dx = ∫_a^c f(x) dx

∫₀¹ [1/(√{4 − x²})] dx + ∫₁² [1/(√{4 − x²})] dx = [(π)/2]

Show that ∫₁² e^x dx = − ∫₂¹ e^x dx

∫₁² e^x dx = e^x |₁²
∫₁² e^x dx = e² − e¹
− ∫₂¹ e^x dx = − e^x |₂¹
− ∫₂¹ e^x dx = −e¹ − (−e²)
− ∫₂¹ e^x dx = e² − e¹

∫₁² e^x dx = − ∫₂¹ e^x dx

Given the values below, use right-hand Riemann Sum with 4 intervals to approximate ∫₀¹⁰ f(x) dx
f(0) = 2 f(3) = 5 f(4) = 5.5 f(7) = 3 f(10) = 7

∫₀¹⁰ f(x) dx ≈ 5(3 − 0) + 5.5 (4 − 3) + 3(7 − 4) + 7(10 − 7)
∫₀¹⁰ f(x) dx ≈ 15 + 5.5 + 9 + 21

∫₀¹⁰ f(x) dx ≈ 50.5

Given the values below, use left-hand Riemann Sum with 4 intervals to approximate ∫₀¹⁰ f(x) dx
f(0) = 2 f(3) = 5 f(4) = 5.5 f(7) = 3 f(10) = 7

∫₀¹⁰ f(x) dx ≈ 2(3 − 0) + 5(4 − 3) + 5.5(7 − 4) + 3(10 − 7)
∫₀¹⁰ f(x) dx ≈ 6 + 5 + 16.5 + 9

∫₀¹⁰ f(x) dx ≈ 36.5

Given the values below, use right-hand Riemann Sum with 2 intervals to approximate ∫₀⁴ f(x) dx
f(0) = 2 f(3) = 5 f(4) = 5.5 f(7) = 3 f(10) = 7

∫₀⁴ f(x) dx ≈ 5(3 − 0) + 5.5(4 − 3)
∫₀⁴ f(x) dx ≈ 15 + 5.5

∫₀⁴ f(x) dx ≈ 20.5

Given the values below, use right-hand Riemann Sum with 5 intervals to approximate ∫₋₅⁵ f(x) dx
f(−5) = 25 f(−1) = 1 f(0) = 0 f(2) = 4 f(4) = 16 f(5) = 25

∫₋₅⁵ f(x) dx ≈ 1(−1 − (−5)) + 0(0 − (−1)) + 4(2 − 0) + 16(4 − 2) + 25(5 − 4)
∫₋₅⁵ f(x) dx ≈ 4 + 0 + 8 + 32 + 25

∫₋₅⁵ f(x) dx ≈ 69

Given the values below, use right-hand Riemann Sum with 3 intervals to approximate ∫₀⁵ f(x) dx
f(−5) = 25 f(−1) = 1 f(0) = 0 f(2) = 4 f(4) = 16 f(5) = 25

∫₀⁵ f(x) dx ≈ 4(2 − 0) + 16(4 − 2) + 25(5 − 4)
∫₀⁵ f(x) dx ≈ 8 + 32 + 25

∫₀⁵ f(x) dx ≈ 65

Given the values below, use left-hand Riemann Sum with 2 intervals to approximate ∫₋₅⁰ f(x) dx
f(−5) = 25 f(−1) = 1 f(0) = 0 f(2) = 4 f(4) = 16 f(5) = 25

∫₋₅⁰ f(x) dx ≈ 25(−1 − (−5)) + 1(0 − (−1))
∫₋₅⁰ f(x) dx ≈ 100 + 1

∫₋₅⁰ f(x) dx ≈ 101

Given the values below, use right-hand Riemann Sum with 4 intervals to approximate ∫₋₂² f(x) dx
f(−2) = −8 f(−1) = −1 f(0) = 0 f(1) = 1 f(2) = 8

∫₋₂² f(x) dx ≈ −1(−1 −(−2)) + 0(0 − (−1)) + 1(1 − 0) 8(2 − 1)
∫₋₂² f(x) dx ≈ −1 + 0 + 1 + 8

∫₋₂² f(x) dx ≈ 8

Given the values below, use left-hand Riemann Sum with 4 intervals to approximate ∫₋₂² f(x) dx
f(−2) = −8 f(−1) = −1 f(0) = 0 f(1) = 1 f(2) = 8

∫₋₂² f(x) dx ≈ −8(−1 − (−2)) + −1(0 − (−1)) + 0(1 − 0) + 1(2 − 1)
∫₋₂² f(x) dx ≈ −8 − 1 + 0 + 1

∫₋₂² f(x) dx ≈ −8

Given the values below, use right-hand Riemann Sum with 6 intervals to approximate ∫₀¹² f(x) dx
f(−2) = −8 f(−1) = −1 f(0) = 0 f(1) = 1 f(2) = 8 f(4) = 64 f(8) = 512 f(10) = 1000 f(12) = 1728

∫₀¹² f(x) dx ≈ 1(1 − 0) + 8(2 − 1) + 64(4 − 2) + 512(8 − 4) + 1000(10 − 8) + 1728(12 − 10)
∫₀¹² f(x) dx ≈ 1 + 8 + 128 + 2048 + 2000 + 3456

∫₀¹² f(x) dx ≈ 7641

Given the values below, use right-hand Riemann Sum with 6 intervals to approximate ∫₀⁸ f(x) dx
f(−2) = −8 f(−1) = −1 f(0) = 0 f(1) = 1 f(2) = 8 f(4) = 64 f(8) = 512 f(10) = 1000 f(12) = 1728

∫₀⁸ f(x) dx ≈ 1(1 − 0) + 8(2 − 1) + 64(4 − 2) + 512(8 − 4)
∫₀⁸ f(x) dx ≈ 1 + 8 + 128 + 2048

∫₀⁸ f(x) dx ≈ 2185

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Riemann Sums, Definite Integrals, Fundamental Theorem of Calculus

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Additional Example 4

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Intro 0:00
Important Equations 0:22

Riemann Sum
Integral
Integrand
Limits of Integration (Upper Limit, Lower Limit)
Other Equations
Fundamental Theorem of Calculus

Lecture Example 1 5:04
Lecture Example 2 10:43
Lecture Example 3 13:52
Additional Example 4
Additional Example 5

College Calculus 1 Online Course

Section 1: Overview of Functions
	Review of Functions	26:29
	Compositions of Functions	12:29
Section 2: Limits
	Average and Instantaneous Rates of Change	20:59
	Limit Investigations	22:37
	Algebraic Evaluation of Limits	28:19
	Formal Definition of a Limit	23:39
	Continuity and the Intermediate Value Theorem	19:09
Section 3: Derivatives, part 1
	Limit Definition of the Derivative	22:52
	The Power Rule	26:01
	The Product Rule	14:54
	The Quotient Rule	19:17
	Applications of Rates of Change	17:43
	Trigonometric Derivatives	26:58
	The Chain Rule	23:47
	Inverse Trigonometric Functions	27:05
	Equation of a Tangent Line	15:52
Section 4: Derivatives, part 2
	Implicit Differentiation	30:05
	Higher Derivatives	13:16
	Logarithmic and Exponential Function Derivatives	17:42
	Hyperbolic Trigonometric Function Derivatives	14:30
	Related Rates	29:05
	Linear Approximation	23:52
Section 5: Application of Derivatives
	Absolute Minima and Maxima	18:57
	Mean Value Theorem and Rolle's Theorem	20:00
	First Derivative Test, Second Derivative Test	27:11
	L'Hopital's Rule	23:09
	Curve Sketching	40:16
	Applied Optimization	25:37
	Newton's Method	25:13
Section 6: Integrals
	Approximating Areas and Distances	36:50
	Riemann Sums, Definite Integrals, Fundamental Theorem of Calculus	22:02
	Substitution Method for Integration	23:19
Section 7: Application of Integrals, part 1
	Area Between Curves	19:59
	Volume by Method of Disks and Washers	24:22
	Volume by Method of Cylindrical Shells	30:29
	Average Value of a Function	16:31
Section 8: Extra
	Graphs of f, f', f''	23:58
	Slope Fields for Differential Equations	18:32
	Separable Differential Equations	17:04

Related Books

	Calculus by Larson, 9th Edition Authors: Ron Larson, Bruce H. Edwards ISBN: 0547167024 Publisher: Brooks Cole Year: 2009 The book is filled with examples and detailed explanations. Capstone exercises are also included for each section and synthesize the main concepts into a single example.
	Calculus by Larson, 9th Edition, Solutions Manual, Volume 1 Author: Ron Larson ISBN: 547213093 Publisher: Brooks Cole Year: 2008 This solutions manual includes worked out solutions to every odd-numbered exercise in Calculus of a Single Variable, 9th edition.
	Calculus by Larson, 9th Edition, Solutions Manual, Volume 2 Authors: Ron Larson, Bruce H. Edwards ISBN: 547213107 Publisher: Brooks Cole Year: 2009 This solutions manual includes worked out solutions to every odd-numbered exercise in Multivariable Calculus, 9th edition.

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