Integration is an operation that finds the **net** area under a curve, and is the opposite operation of differentiation. As such, it is also known as **anti-differentiation**.
The area under a curve between the interval of x-values $[a,b]$ is:
$$A=\lim_{x\to\infty}\sum^n_{i=1}f(x_i)\Delta x$$
which can be simplified to, where $dx$ indicates that integration should be performed with respect to $x$:
$$A=\int^b_a f(x)dx$$
While $\Sigma$ refers to a finite sum, $\int$ refers to the sum of a limit.
As integration is the opposite operation of differentiation, they can cancel each other out.
$$\frac{d}{dx}\int f(x)dx=f(x)$$
The **integral** or **anti-derivative** of a function is capitalised by convention. Where $C$ is an unknown constant:
$$\int f(x)dx=F(x)+C$$
When integrating, there is always an unknown constant $C$ as there are infinitely many possible functions that have the same rate of change but have different vertical translations.
!!! definition
- $C$ is known as the **constant of integration**.
- $f(x)$ is the **integrand**.
### Integration rules
$$
\begin{align*}
&\int 1dx &= &&x+C \\
&\int (ax^n)dx, n≠-1 &=&&\frac{a}{n+1}x^{n+1} + C \\
&\int (x^{-1})dx&=&&\ln|x|+C \\
&\int (ax+b)^{-1}dx&=&&\frac{\ln|ax+b|}{a}+C \\
&\int (ae^{kx})dx &= &&\frac{a}{k}e^{kx} + C \\
&\int (\sin kx)dx &= &&\frac{-\cos kx}{k}+C \\
&\int (\cos kx)dx &= &&\frac{\sin kx}{k}+C \\
\end{align*}
$$
Similar to differentiation, integration allows for constant multiples to be brought out and terms to be considered individually.
Similar to limit evaluation, the substitution of complex expressions involving $x$ and $dx$ with $u$ and $du$ is generally used to work with the chain rule.
Regions **under** the x-axis are treated as negative while those above are positive, cancelling each other out, so the definite integral finds something like the net area over an interval.
If $f(x)$ is continuous at $[a,b]$ and $F(x)$ is the anti-derivative, the definite integral is equal to:
$$\int^b_a f(x)dx=F(x)\biggr]^b_a=F(b)-F(a)$$
As such, it can be evaluated manually by integrating the function and subtracting the two anti-derivatives.
!!! warning
If $u$-substitution is used, the limits of integration must be adjusted accordingly.
To find the total **area** enclosed between the x-axis, $x=a$, $x=b$, and $f(x)$, the function needs to be split at each x-intercept and the absolute value of each definite integral in those intervals summed.
$$A=\int^b_a \big|f(x)\big| dx$$
### Properties of definite integration
The following rules only apply while $f(x)$ and $g(x)$ are continuous in the interval $[a,b]$ and $c$ is a constant.
To find the area enclosed between two curves, the graph should be sketched if possible and their points of intersection determined to identify which parts of each function are on top of the other at any given time. An interval chart may be helpful. For each section, where $f(x)$ is always greater than $g(x)$ in the interval $[a,b]$:
$$A=\int^b_a [f(x)-g(x)]dx, f(x)\geq g(x)\text{ in } [a,b]$$
If the limits of integration are not given, they are the outermost points of intersection of the two curves.
Shapes formed by rotating a line or curve about a fixed axis, such as cones, spheres, and cylinders are all known as **solids of revolution**. By splicing each shape into infinitely small disks, the cylinder volume formula can be used to find the volume of the solid.
Events $A$ and $B$ are **disjoint** or mutually exclusive if no outcomes between them are common and can never happen simultaneously. As such the probability of one of the events happening is equal to their sum.
$$
P(A\cup B)=P(A)+P(B) \\
P(A\cap B)=0
$$
Events $A$ and $B$ are **exhaustive** if their union includes all possible outcomes in the sample space: $A\cup B=U$.
### Probability distributions and discrete random variables
The **discrete random variable**, $X$, represents a **quantifiable**, measurable, discrete quantity. The lowercase $x$ represents a possible value of $X$.
The probability that $X$ takes on any one of the specific possible outcomes is written as $P(X=x)$. The sum of the probability all possible outcomes must still remain $1$:
$$\Sigma P(X=x)=1$$
!!! example
In an experiment of tossing a coin twice, possible values of $X$ include $0,1,2$ so $x\in\{0, 1, 2\}$.
A **probability distribution** is a distribution of outcomes and their probabilities. Events/outcomes are placed on the top row while probability is provided on the bottom in the form of a fraction. Probability distributions can also be graphed with the outcomes on the x-axis and their probabilities on the y-axis with lines similar to a bar graph sitting on the grid lines to represent a probability..
!!! example
For the coin ross experiment in the previous example, where $X$ is the number of tails when tossing a coin twice:
The **expected value** of an experiment or the "expectation of $X$" is the mean value of $X$ that is expected to be obtained over many trials. It is equal to the sum of the value of all outcomes multiplied by their probability.
$$
\begin{align*}
E(X)&=\Sigma P(X=x)x \\
&=\mu=x_1p_1+x_2p_2+...+x_kp_k
\end{align*}
$$
!!! warning
It is possible that the expected value will not be a value in the set, and the expected value should **not be mistaken** with the outcome with the highest probability.
**Bernoulli trials** have a fixed number of trials that are independent of each other and identical with only two possible outcomes — a success or failure.
Where $r$ is the number of successes in a Bernoulli trial:
$$P(X=r)={n\choose r}p^rq^{n-r}$$
where ${n\choose r}=\frac{n!}{r!(n-r)!}$
A binomial distribution is a probability distribution of two possible events, a success or a failure. The distribution is defined by the number of trials, $n$, and the probability of a success, $p$. The probability of failure is defined as $q=1-p$.
$X\sim$ denotes that the random variable $X$ is distributed in a certain way. Therefore, the binomial distribution of $X$ is expressed as:
$$X\sim B(n, p)$$
In a binomial distribution, the expected value and **variance** are as follows:
$$
E(X)=np \\
Var(X)=npq
$$
On a graphing display calculator, where $r$ is the number of successes:
Also known as **Gaussian distribution** or in its graphical form, a normal or bell curve, the normal distribution is a **continuous** probability distribution for the random variable $x$.
- The normal curve is bell-shaped and symmetric about the mean.
- The area under the curve is equal to one.
- The normal curve approaches but does not touch the x-axis as it approaches $\pm \infty$.
From $\mu-\sigma$ to $\mu+\sigma$, the curve curves downward. $\mu\pm\sigma$ are the **inflection points** of the graph. It is expressed graphically as:
~68%, ~95%, and ~99.7% of the data is found within one, two, and three standard deviations of the mean, respectively.
### Standard normal distribution
The **standard normal distribution** has a mean of 0 and standard deviation of 1. The horizontal scale of the standard normal curve corresponds to **$z$-scores** that represent the number of standard deviations away from the mean. To convert an $x$-score to a $z$-score:
$$z=\frac{x-\mu}{\sigma}$$
A **Standard Normal Table** can be used to determine the cumulative area under the standard normal curve to the left of scores -3.49 to 3.49. The area to the *right* of the score is equal to $1-z_\text{left}$. The area *between* two z-scores is the difference in between the area of the two z-scores.
To standardise a normal random variable, it should be converted from the form $X\sim N(\mu,\sigma^2)$ to $Z\sim N(0,1)$ via the formula to convert between x- and z-scores.
The probability of a z-score being less than a value can be rewritten as phi.
$$P(z<a)=\phi(a)$$
Some z-score rules partially taken from probability rules:
$$
\begin{align*}
P(z>-a)&=P(z<a) \\
1-P(z>a)&=P(z<a)
\end{align*}
$$
On a graphing display calculator:
The `normalcdf` command can be used to find the cumulative probabilty in a normal distribution in the format $\text{normalcdf}(a,b,\mu,\sigma)$, which will solve for $P(a<x<b)$.$-1000$isgenerallyasufficientlylowvaluetosolveforjust$P(x<b)$.
Please see [SL Physics 1#1.3 - Vectors and Scalars](/sph3u7/#13-vectors-and-scalars) for more information.
One vector can be represented in a variety of methods. The algebraic form $(1, 2)$ can also be represented in the alternate algebraic forms $[1, 2]$ and $1\choose 2$.
Where $v$ is the vector, $A$ is the initial and $B$ is the terminal point of the vector, a vector can be identified by any of the following symbols:
- $\vec{AB}$
- $\vec{v}$
- $\boldsymbol{v}$ (bolded)
The special **zero vector** $\vec{0}$ is a vector of undefined direction and zero magnitude.
Vectors with the same magnitude but opposite directions are equal to one another except one is the negative of the other.
**Colinear** vectors are those that parallel with one another — that is, with identical or opposite directions. Vectors that are colinear must also be **scalar multiples** of each other:
$$\vec{u}=k\vec{v}$$
**Position** vectors are vectors where the initial point is at the origin — where the terminal point is $A$, a position vector can be written as $\vec{OA}$.
**Colinear points** are points that lie on the same straight line. If two colinear vectors that share a common point can be formed between three points, those points are colinear.
