# Calculus with analytic geometry II

## Contents

1 Integration by parts

2 Trigonometric integrals

3 Trigonometric substitution

4 Partial fractions

5 Summary of integration techniques

6 Improper integrals

7 Separable differential equations

8 Approximate methods for integrals and differential equations

9 Arc length and area of surfaces of revolution

10 Applications to physics

11 Sequences

12 Series

13 The integral convergence test

14 Comparison tests

15 Alternating series

16 Absolute convergence, root and ratio tests

17 Definition and basic properties of power series

18 Representing functions by power series

Bonus: A relation between power series and differential equations

19 Taylor and Maclaurin series

20 Approximating functions with Taylor polynomials

21 Parametric curves

22 Polar coordinates and applications

0.0 Welcome video

## Integration by parts

1 Notes on integration by parts
1.1 Integration by parts: motivation and formula
1.2 Integrating $$x \sin x$$ and the choice of $$u$$
1.3 How to not integrate $$x \sin x$$

Do not repeat at home
1.4 Getting rid of the second power of $$t$$ by integrating by parts

Do repeat at home (twice)
1.5 Integrating inverse functions and definite integrals by parts
1.6 Integration by parts with moving terms to the other side

In this example, we repeatedly integrate by parts and move some of the terms around to arrive at the answer
1.7 We obtain the reduction formula for $$\sin x$$

This example shows how trig identities will be useful for integration, and prepares us for the next section

## Trigonometric integrals

2 Notes on trigonometric integrals
2.1 What is a trigonometric integral
2.2 We integrate $$\sin^5 x \, \cos^2 x$$

What is common to these products so far?
2.3 Applying half-angle formulas

After realizing that the previous examples contain odd degrees of $$\sin, \cos$$, we discuss dealing with even powers
2.4 In order to integrate $$\sin^4 x$$, half-angle formulas have to be applied repeatedly
2.5 The above examples are summarized into a general strategy for integrating products of $$\sin x$$ and $$\cos x$$
2.6 What is essential about our integration strategy? Why does it work?
2.7 Examples of integrating products of $$\tan x$$ and $$\sec x$$
2.8 A general strategy for products of $$\tan x$$ and $$\sec x$$ is obtained in this video
2.9 Some special cases not covered by the strategy for $$\tan x$$ and $$\sec x$$

Some products of $$\tan x$$ and $$\sec x$$ are still not covered by our general strategy. This video discusses how to use the Pythagorean identity and integration by parts to obtain reduction formulas for such cases, not covered previously.

## Trigonometric substitution

3 Notes on trigonometric substitution
3.1 Motivation for trig sub

Evidence that trig sub shows up in some natural integrals.
3.2 Main table of trigonometric substitutions

We summarize the motivational examples into a table of different trig subs and identities that explain them.
3.3 Substituting $$x = a \sin\theta$$
3.4 Substituting $$x = a \sec\theta$$
3.5 Definite integrals with trig sub and handling coefficients on $$x^2$$

Instead of $$\pm x^2 \pm a^2$$ under the radical, in this example we have $$\pm c^2 x^2 \pm a^2$$. We show how to factor out the coefficient at $$x^2$$ from the integral.
3.6 Completing the square to apply a trig sub

In this example, expression under the square root is not of the form $$\pm x^2 \pm a^2$$, but instead a complete quadratic polynomial. We show how to reduce it to the standard form.

## Partial fractions

4 Notes on partial fraction decomposition
4.1 Motivation for partial fraction decomposition

Definition of a rational function and the general idea of partial fraction decomposition
4.2 Main table of partial fraction decomposition

An overview of the algorithm for partial fraction decomposition and a table for the partial fractions corresponding to different irreducible factors in denominator.
4.3 Long division and unique linear factors

We use long division to split an improper rational function into a sum of a polynomial with a proper part. In the second example, we integrate a rational function with unique (nonrepeated) factors in the denominator.
4.4 Repeated linear factors

An example of a rational function with repeated linear factors in the denominator. We determine the coefficient by equating the coefficients for the powers of $$x$$.

Integration of a rational function with unique (nonrepeated) irreducible quadratic terms.

Integration of functions with repeated irreducible quadratic terms. In particular, we see how to apply trig sub to negative powers of $$1+x^2$$.
4.7 Completing a square in a partial fraction

We demonstrate how to integrate a partial fraction with an irreducible quadratic factor by completing a square, and applying a u-sub.
4.8 Rationalizing substitution

When integrating an expression with an n-th order root, we introduce a u-sub equal to this root, and reduce the problem to integration of a rational function.

## Summary of integration techniques

5 Notes on the integration strategy
5.1 An overview of integration heuristics
5.2 Examples 1-2

These examples illustrate simplification of the integrand before integration and combining two different techniques: u-sub and integration by parts.
5.3 Examples 3-5

A discussion of the PFD and the usefulness of algebraic transformations, such as multiplying by the conjugate expression.
5.4 Conclusion: not all expressions have antiderivatives in elementary functions

Our course just scratches the surface of the vast functional universe. The antiderivatives of some of its inhabitants can only be expressed through special functions (error function, Bessel functions, etc), others don’t have a nice representation at all.

## Improper integrals

6 Notes on improper integrals
6.1 Definition of an improper integral
6.2 Convergence of the integral of $$1/x^p$$ for different values of $$p$$
6.3 Using integration by parts in improper integrals
6.4 Improper integrals of type II and the comparison theorem

Improper integrals of the second type contain a function that becomes infinite at some point of the integration interval. We also discuss the comparison theorem and see its application to integral of $$e^{-x^2}$$. Specifically, even though this function does not have an antiderivative in elementary functions, we are able to determine that its improper integral converges.

## Separable differential equations

7 Notes on differential equations
7.1 Introduction: why we care about integration techniques and an outline of the solution process
7.2 Applying the initial condition to select a specific solution
7.3 When solving for $$y$$ can be tricky
7.4 Absorb $$\pm$$ into the positive constant, by extending the range of admissible constants

This is useful when solutions have the form $$|y| = K e^x, \ K >0$$.
7.5 Modeling a simple electric circuit
7.6 Mixing problems

A common class of problems which involve uniform mixing of a solution.
7.7 Orthogonal trajectories

These generalize both Cartesian and polar coordinates. Imagine curved coordinate lines!

## Approximate methods for integrals and differential equations

8 Notes on approximate methods

8.1 Introduction: when it is necessary to solve an integral or a differential equation approximately
8.2 Derivation of the midpoint and trapezoidal rules

These rules are obtained by approximating area under the curve with rectangles and trapezoids, respectively.
8.3 Trapezoidal vs midpoint rule: which is better?

See if you favorite is a more precise rule!
8.4 Error bounds for $$T_n$$ and $$M_n$$
8.5 Simpson’s rule and its error bound
8.6 Numerical integration in SageMath
8.7 Euler’s method

This video discusses how to view a differential equation as a vector field, by attaching a little vector to every point of the $$x,y$$ plane, and how to construct approximate solutions by connecting the points obtained from Euler’s method.
8.8 Euler’s method in SageMath

## Arc length and area of surfaces of revolution

9 Notes on arc length
9.1 Derivation of the formula for the arc length of a curve
9.2 First example: length of the curve $$y^2 = x^3$$
9.3 Treating a curve as the graph of $$x = x(y)$$
9.4 The arc length function: definition and use
9.5 Summary of arc length
Notes on areas of surfaces of revolution
9.6 Derivation of the formula for the area of surface of revolution
9.7 Area of a spherical slice and using cylindrical projection in cartography
9.8 Rotation of an arc of $$y=x^2$$.

Solved two ways: integrating in $$x$$ and in $$y$$. The variable of integration is independent of the axis.
9.9 Summary: how to set up the integral for a surface of revolution

## Applications to physics

10 Notes on applications to physics
10.1 Hydrostatic force: derivation and example
10.2 Center of mass of a region: formula and example

## Sequences

11 Notes on sequences
11.1 Definition of a sequence and some generic examples: populations, digits of $$e$$, etc.
11.2 Limits of sequences and limit laws
11.3 Lifting to functions: to apply l’Hôpital’s rule, pass to the function first!
11.4 Interchanging limits with continuous functions
11.5 When is the sequence $$r^n$$ convergent?

This will be useful later, when we discuss the geometric series.
11.6 Monotonic sequences

We discuss increasing and decreasing sequences, and a theorem about them. This theorem will be used for establishing convergence of series with nonnegative terms.

## Series

12 Notes on series
12.1 Definition of a series and its sum

Any decimal expansion is an example of number series!
12.2 The first example; finding $$a_k$$ through $$s_n$$
12.3 Telescoping series
12.4 Geometric series: condition of convergence and sum formula
12.5 Examples of geometric series

How to recognize a geometric series and evaluate its sum
12.6 Rewriting a number with periodic decimal expansion as a fraction

If the decimal expansion of a number is periodic, it can be represented as an irreducible fraction. In this example, we show how. (In contrast, say, $$\sqrt 2$$ or $$\pi$$ have decimal expansions that are not periodic.)
12.7 Test for divergence and the harmonic series

If the terms of a series $$a_k$$ do not converge to 0, this series is divergent (does not have a finite sum). The converse is not true: if $$a_k \to 0,\ k\to \infty$$, their sum $$\sum_k a_k$$ is not necessarily convergent. An example of such series is $$\sum_k \frac 1k$$. Clearly $$1/k \to 0,\ k\to \infty$$, and yet the series diverges.
12.8 A short illustration of how the test for divergence is applied
12.9 Arithmetic operations with series

The limit laws enable adding limits and multiplying them by constants. Because the sum of a series is the limit of its partial sums, the limit laws for addition and subtraction extend to series.
12.10 A glimpse of power series

As a sneak peek into what we will discuss next, we obtain an expansion of $$\frac1{1-x}$$ into a power series. 😱

## Integral test

13 Notes on the integral test
13.1 Showing that $$\sum_{k} \frac1{k^2}$$ is convergent, using integrals
13.2 Showing that $$\sum_k \frac1{k^{1/2}}$$ is divergent and the statement of the integral test
13.3 Convergence of $$\sum_k \frac1{k^{2}+1}$$
13.4 For which $$p$$ is $$\sum_k \frac1{k^{p}}$$ convergent? (For $$p>1$$, just as improper integrals.)
13.5 Convergence of $$\sum_k \frac{\ln k}{k}$$

In order to show that $$f(x) = \ln(x)/x$$ is decreasing, we have to compute its derivative. It turns out, the function is decreasing only for $$x>e$$, but we can still apply the integral test.
13.6 Formulation of the remainder estimate

Using the ideas from integral test, we estimate the n-th remainder, $$R_n = s - s_n$$ for a series $$\sum_k a_k$$, converging to the finite value $$s$$.
13.7 Applying the remainder estimate to the series $$\sum_k \frac1{k^3}$$

## Comparison tests

14 Notes on the comparison tests
14.1 The statement of the comparison test
14.2 Applying the comparison test: compare a given series to geometric and p-series
14.3 Limit comparison test

If obtaining an inequality comparing $$a_k$$ and $$b_k$$ is difficult, it is useful to consider the limit of their ratio: $$\lim_{k\to\infty} \frac{a_k}{b_k}$$

## Alternating series

15 Notes on alternating series
15.1 Definition of alternating series and statement of the alternating series test
15.2 Applying the alternating series test

## Absolute convergence, root and ratio tests

16 Notes on absolute convergence, root and ratio tests
16.1 The notion of absolute convergence

Absolute convergence allows to bypass the specifics of signs of $$a_k$$ in a series $$\sum_k a_k$$, and only look at the series of absolute values $$\sum_k |a_k|$$. Then convergence of this series of absolute values implies usual convergence of the original series $$\sum_k a_k$$.
16.2 Application of absolute convergence to the series $$\sum_k \frac{\cos k}{k^2}$$
16.3 The ratio test: statement

Especially suitable for expressions containing e.g. $$a^k$$ and $$k!$$.
16.4 Ratio test applied
16.5 The root test: statement and an example

Especially suitable for $$a_k = (some\ expression)^k$$.

## Power series

17 Notes on power series
17.1 Definition of power series

The image quality is not great, and the colors are inverted. This has been corrected in subsequent videos
17.2 Find the values of $$x$$ that make the power series $$\sum_k \frac{(x-3)^k}k$$ converge
17.3 Finding the regions of convergence for $$\sum_k k!x^k$$ and the Bessel function $$J_0$$
17.4 We explore the graph of the Bessel function $$J_0$$
17.5 Summary of different convergence modes for power series.
17.6 Using the theorem from 1.5, we determine the interval of convergence (= all the values of $$x$$ for which it converges) for the series $$\sum_k \frac{(-3)^k x^k}{\sqrt{k+1}}$$.

## Representing functions by power series

18 Notes on representing functions by power series
18.1 Representing functions with power series using algebraic manipulations
18.2 The main theorem concerning differentiation and integration of power series, and an example that involves differentiating a series
18.3 A comment on changing indexing in a series
18.4 Two important examples: power series for $$\ln(1+x)$$ and $$\arctan x$$
18.5 Discussion of the series for $$\ln(1+x)$$ and $$\arctan x$$
18.6 Representing an antiderivative of a function by the power series, and why it is useful

## Bonus: A relation between power series and differential equations

A relation between power series and differential equations.
Connecting two major topics of this course, we solve the homework question #8 from section 11.9. First, we show that the power series $$f(x) = \sum_{n=0}^\infty \frac{x^n}{n!}$$ satisfies the differential equation $$f’(x)=f(x)$$. Then, we solve this differential equation and verify that its unique solution with the initial condition $$f(0)=1$$ is $$e^x$$. This gives an alternative way to compute the power series representation for $$e^x$$.

## Taylor and Maclaurin series

19 Notes on Taylor and Maclaurin series
19.1 The main theorem about Taylor series is discussed, which gives the expressions for the coefficients in the Taylor series of a function through its derivatives
19.2 Computing the Maclaurin series for $$e^x$$
19.3 How to prove that Taylor series of a function converges to it? We discuss the remainder estimate
19.4 Using the remainder estimate, we show that the Maclaurin series for $$e^x$$ converges to it
19.5 Maclaurin series for $$\sin x$$
19.6 Using the previous example, we calculate the Maclaurin expansion for $$\cos x$$
19.7 We apply algebraic manipulations to the series for $$\cos x$$ in order to compute the Maclaurin series of $$x \cos(x^3)$$ and its radius of convergence
19.8 The binomial series: expansion of $$(1+x)^p$$

For more context about binomial coefficients arising in polynomials, read this note.
19.9 Proof of the main theorem about Taylor series from 19.1
Notes on examples and applications of Taylor and Maclaurin series
19.10 We consider an application of the binomial series formula, obtaining the Maclaurin expansion for $$\frac1{\sqrt{4-x}}$$
19.11 An example of computing a Taylor series — the expansion of $$\sin x$$ around the center $$a = \frac{\pi}{3}$$
19.12 Using power series to compute limits

Turns out, not only power series can be used to approximate functions and compute antiderivatives, you can also use them instead of l’Hôpital’s rule to evaluate limits!
19.13 Example of multiplying two power series

We mentioned earlier that power series were the closest thing to infinite polynomials. In this video we show they are in fact so close to polynomials, one can multiply two power series together!
19.14 Division of power series

If the previous video was somewhat surprising, this one is nothing short of astounding. We will perform long division on infinite series, just like as if they were polynomials 😮

## Approximating functions with Taylor polynomials

20 Notes on approximation with Taylor polynomials
20.1 Approximating $$\sqrt[3]x$$ close to the point $$a = 8$$ with $$T_2(x)$$, and estimating the resulting error

We discuss a typical application of Taylor series — approximation of a function close to a point where its value is known. This is how Taylor series usually show up in computational applications.
20.2 Antiderivative of $$e^{-x^2}$$

The antiderivative of $$e^{-x^2}$$ cannot be expressed in terms of elementary functions. It can, however, be represented by a (Maclaurin) power series converging everywhere. We obtain such representation, demonstrating the expressiveness of power series.
20.3 Comparison of the initial Taylor polynomials

We plot the function from episode 20.2 alongside several of its Taylor polynomials and discuss the code of the Python routine used to produce these graphs.

## Parametric curves

21 Notes on parametric curves
21.1 The definition of the parametric curve. Curve as a trajectory of motion
21.2 Parameter elimination technique

We introduce the primary technique for expressing a parametric curve as a regular curve: by eliminating the parameter $$t$$. This technique is used to identify a parametric curve.
21.3 The parametric curve $$(x(t),\, y(t)) = (t^2, \ln t)$$

We show that every parametric curve has two characteristics: the shape and the direction of motion of a particle along this shape. Plus, in this example we get to reverse time Dr. Who-style 😮
21.4 The parametric curve $$(x(t),\, y(t)) = (\sin t, \sin^2 t)$$

A piece of parabola along which we oscillate back and forth
21.5 The parametric curve $$(x(t),\, y(t)) = (\tan^2\theta, \sec \theta )$$, which shows that the range of the parameter is important
21.6 Tangents to parametric curve

We look at a self-intersecting curve, which has not one, but two tangents at the same point! (and does not pass the vertical line test)
21.7 Computing arc length of parametric curves

The formula and an example: length of a circle
21.8 Area of surfaces of revolution for parametric curves

The formula and an example: area of a sphere

## Polar coordinates and applications

22.8 An example for area in polar coordinates: the region inside the circle $$r = 3\sin \theta$$ and outside the cardioid $$r = 1+\sin\theta$$