Calc 2: Integrals & Sums

We'll wrestle with reversing the derivative, then explore the behavior of sums and their connection to functions!

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