NotesAtHKU

Definite integrals

Signed areas

Signed area is the area between the curve and the x-axis, where the area above the x-axis is positive and below is negative.

Properties of definite intergrals

$\int_{a}^{b}f(x)dx=-\int_{b}^{a}f(x)dx$
$\int_{a}^{a}f(x)dx=0$
$\int_{a}^{b}f(x)dx+\int_{b}^{c}f(x)dx=\int_{a}^{c}f(x)dx$
$\int_{a}^{b}kf(x)dx=k\int_{a}^{b}f(x)dx$
$\int_{a}^{b}(f(x)\pm g(x))dx=\int_{a}^{b}f(x)dx\pm\int_{a}^{b}g(x)dx$

Fundamental theorem of calculus

Fundamental theorem of calculus (FTC)

If $f(x)$ is continuous in $[a,b]$ , then:

\int_{a}^{b}f(x)dx=F(b)-F(a)

Where $F(x)$ is the definite integral of $f(x)$ .

Second fundamental theorem of calculus

If $f(x)$ is continuous in $[a,b]$ , then:

\frac{d}{dx}\int_{a}^{x}f(t)dt=f(x)

Find $g'(x)$ for $g(x)=\int_1^{x^2}\cos t\ dt$ :

We first define $G(u)=\int_1^{u}\cos t\ dt$ , then we can apply the chain rule:

\begin{aligned} g'(x) & =(G(x^2))' \&=G'(x^2)\cdot2x\&=\cos x^2\cdot2x \end{aligned}

Cases of area 0

For $\int^{b}_{a}f(x)dx$ :

If $b=a$ , then the area is 0 (no width)
If $f(x)=f(-x)$ and $a=-b$ , then the area is 0 (symmetry)

Limit of Riemann sums

We can convert a limit to infinity of a Riemann sum to a definite integral:

\lim_{n\to\infty}\sum_{i=1}^{n}f(a+i\Delta x)\Delta x=\int_{0}^{b}f(a+x)dx,\quad\Delta x=\frac{b-a}{n}

Some properties of summation:

$x\sum f(x) = \sum x f(x)$
$\sum_{i=1+n} f(x, i), \text{let }k=i-n\to\sum_{k=1}^{n}f(x, k+n)$

Techniques of integration

Overview

By substitution: “sub $g(x)\to u$ , find $du$ and replace all functions of $x(dx)$ with $u(du)$ ”
By part: “ $\int ab=a\int b-\int (a'\times \int b)$ ”
By joining recurring parts: “ $\int ab=f(ab)+\int ab\to 2\int ab = f(ab)$ ”
By partial fractions: “ $\frac{k}{f(x)g(x)}=\frac{A}{f(x)}+\frac{B}{g(x)}$ ”
By trigonometric identities
By multiplying fractions by one: “Multiplying $\frac{\sec^2x}{\sec^2x}$ to fit trigo form”

Integration by substitution

For $\int f(g)g'\:dx$ :

Let $u=g$
Find $du=g'dx$
Change limits in terms of $u$ if definite
Replace all $g\to u$ and $g'dx\to du$
Integrate, then subsitute $u\to g$

Integration by parts

\int ab=a\int b-\int (a'\times \int b)

Tips: Let $a$ (differentiating term) to the first term you see on the list:

Logarithmic
Inverse trigo
Alebratic (polynomial)
Trigo
Exponential

Deriving the IBP formula

The formula is derived from the product rule of differentiation:

\begin{aligned} (ab)' & = a'b + b'a \\ b'a & = (ab)' - a'b \\ \int b'a & = ab - \int a'b \\ \int ab & =a\int b-\int (a'\times \int b) \end{aligned}

Trigo-identities substitution tips

Use the following trigonometric identities to simplify the integral:

$\sin^2x+\cos^2x=1$
$\sin^2x=\frac{1-\cos2x}{2}$
$\cos^2x=\frac{1+\cos2x}{2}$
$2\sin x\cos x=\sin2x$
$\cos2x=\cos^2x-\sin^2x$
$\sec^2x=1+\tan^2x$
$\csc^2x=1+\cot^2x$

Use the following substitutions for similar expressions:

\text{Expressions of the form } \begin{cases} a^2-f(x)^2 & \to f(x)=a\sin\theta \\ a^2+f(x)^2 & \to f(x)=a\tan\theta \\ f(x)^2-a^2 & \to f(x)=a\sec\theta \\ \end{cases}

Remember that this uses integration by substitution, so we need to find $dx$ in terms of $\theta$ .

Solids of revolution

A 3D shape formed by rotating a region around an axis.

A region is defined by 2 curves and an interval:

The vertical region bounded by an outer cruve $R(x)$ and an inner curve $r(x)$ .
The inner curve is $y=0$ for the region bounded by the $x$ -axis.
The horizontal region is bounded by an interval $x:\ [a, b]$

Note that for the methods below, the terms $x$ and $y$ can be switched to fit the problem.

$\{f(x)\to\ \circlearrowright x\}$ denotes using a function of $x$ to find the volume of a solid of revolution around the $x$ -axis.

If the volume is generated by the rotation about an axis other than the main axes, we can simply shift the function to fit the main axes.
If the curves in the given range does not form a closed region with only acute angles, we can split the region into multiple parts.

Volume by Washers (Disk)

\{f(x)\to\ \circlearrowright x\}:\quad V=\pi\int_{a}^{b}R^2(x)-r^2(x)dx

For rotating about $y=n$ : $\forall R/r(x)\to n-R/r(x)$

Volume by Cylindrical shells

\{f(x)\to\ \circlearrowright y\}:\quad V=2\pi\int_{a}^{b}[x][R(x)-r(x)]dx

For rotating about $x=n$ : $[x]\to [x-n]$

Arcs and surfaces

Arc length

The arc length of a curve $y=f(x)$ from $x=a$ to $x=b$ is given by:

L=\int_{a}^{b}\sqrt{1+f'^2(x)}dx

And for parametized equations $x=f(t)$ , $y=g(t)$ :

L=\int_{a}^{b}\sqrt{f'^2(t)+g'^2(t)}dt

For polar equations $r=f(\theta)$ :

L=\int_{a}^{b}\sqrt{r^2+r'^2(\theta)}d\theta

Area surfaces of revolution

Used to find the surface area generated by rotating a curve along an axis.

\{f(x)\to\ \circlearrowright x\}:\quad A=2\pi\int_{a}^{b}f(x)\sqrt{1+f'^2(x)}dx

And for parametized equations $x=f(t)$ , $y=g(t)$ :

A=2\pi\int_{a}^{b}g(t)\sqrt{f'^2(t)+g'^2(t)}dt

Integrals

Definite integrals

Signed areas

Properties of definite intergrals

Fundamental theorem of calculus

Fundamental theorem of calculus (FTC)

Second fundamental theorem of calculus

Cases of area 0

Limit of Riemann sums

Techniques of integration

Overview

Integration by substitution

Integration by parts

Deriving the IBP formula

Trigo-identities substitution tips

Solids of revolution

Solids of revolution

Volume by Washers (Disk)

Volume by Cylindrical shells

Arcs and surfaces

Arc length

Area surfaces of revolution

On this page