Calc 2 Integrals & Sums

What you'll learn
Exploring the connection between reversing a derivative ("integration") and area under the curve
Developing problem solving techniques for integration
Understanding the connection between limits, sequences, and sums of sequences ("series")
Developing problem solving techniques for determining whether a series converges
Extending series to include a variable, primarily to enable approximating common functions
Requirements
Experience with algebra, trigonometry, and calculus 1 — take a look at our other courses if you need a refresher!
Description
A typical calculus 2 course covers two main topics: integrals and sums.Integrals describe the area under a curve over a specific interval, and are the inverse operation of a derivative. Because it's not tied to a specific approach (unlike derivatives, which are generated by the difference quotient), we'll get to explore the puzzle-like world of integration. You'll learn how to think about questions like:What is the relationship between areas, antiderivatives, and the values input into these?What tools do we have based on our known derivatives? What about based on our known derivative rules?Is the integral a linear operator — meaning sums & constant multiples can be "pulled apart" — like its inverse can?What can we do with a function once we are able to integrate it?Sums take a list of terms (a sequence) and add them all up. The most interesting property of a sum is whether or it converges (eventually sums to a specific number) or diverges (doesn't). Then, with some clever calculus, we can use these series to approximate more complex functions that are hard to evaluate otherwise! You'll learn how to think about questions like:What is convergence, and how do I get a "feel" for whether a series converges or diverges?How can we use properties or other series to help us identify convergence for series we don't already know?How can we generate approximations for functions as sums of polynomials?What ways can we extend these approximations, and how accurate are they?Can't wait to get started! Calculus 2 is an exciting opportunity to flex your creative muscles in a field that just gets more creative the deeper you dive in. Let's go!
Overview
Section 1: Introduction
Lecture 1 Welcome to Calc 2: Integrals & Sums!
Lecture 2 Prerequisites in More Detail
Lecture 3 Big Picture Idea: Integrals
Lecture 4 Big Picture Idea: Sums
Lecture 5 Getting Ready
Section 2: Integral Concepts
Lecture 6 What is a Riemann Sum?
Lecture 7 Upgraded Area Methods
Lecture 8 Fundamental Theorem of Calculus
Lecture 9 Power Rule for Integrals
Lecture 10 Trig Integrals, a First Pass
Lecture 11 Exponential & Harmonic Integrals
Lecture 12 Inverse Trig Integrals, a First Pass
Section 3: Integration Techniques
Lecture 13 u Substitution
Lecture 14 Parts
Lecture 15 Partial Fractions
Lecture 16 Trig Substitution
Section 4: Integral Applications
Lecture 17 Average Value of a Function
Lecture 18 Area Between Curves
Lecture 19 Volumes by Revolution
Lecture 20 Reducing to One Variable
Section 5: Series Convergence
Lecture 21 Sequences
Lecture 22 What is Convergence?
Lecture 23 Geometric Series
Lecture 24 Integral Test
Lecture 25 Direct Comparison Test
Lecture 26 Limit Comparison Test
Lecture 27 Alternating Series Test
Lecture 28 Test for Divergence
Lecture 29 Root & Ratio Tests
Lecture 30 Telescoping Series
Section 6: Power Series
Lecture 31 What is a Power Series?
Lecture 32 Taylor Series
Lecture 33 Maclaurin Series
Lecture 34 Series Error Bounds
Lecture 35 Calculus of Power Series
Section 7: Practice Set A
Lecture 36 Set A Introduction
Lecture 37 Problem 1
Lecture 38 Problem 2
Lecture 39 Problem 3
Lecture 40 Problem 4
Lecture 41 Problem 5
Lecture 42 Problem 6
Lecture 43 Problem 7
Lecture 44 Problem 8
Lecture 45 Problem 9
Lecture 46 Problem 10
Lecture 47 Problem 11
Lecture 48 Problem 12
Lecture 49 Problem 13
Lecture 50 Problem 14
Lecture 51 Problem 15
Lecture 52 Problem 16
Lecture 53 Problem 17
Lecture 54 Problem 18
Section 8: Bonus Lecture
Lecture 55 Bonus Lecture: UpGrade University
Lovers and future lovers of math who are ready to learn by exploring concepts,Calc 2 is a common early college course for technical fields of study, and is a big step toward a more creative understanding of math
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