8.1 KiB
SL Math - Analysis and Approaches - 2
The course code for this page is MCV4U7.
Integration
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.
\[ \begin{align*} &\int k\cdot f(x)dx&=&&k\int f(x)dx \\ &\int[f(x)\pm g(x)]dx&=&&\int f(x)dx \pm \int g(x)dx \end{align*} \]
Indefinite integration
The indefinite integral of a function contains every possible anti-derivative — that is, it contains the constant of integration \(C\). \[\int f(x)dx=F(x)+C\]
Substitution rule
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. \[ u=g(x) \\ \int f(g(x))\cdot g´(x)\cdot dx = \int f(u)\cdot du \]
??? example To solve \(\int (x\sqrt{x-1})dx\): \[ let\ u=x-1 \\ ∴ \frac{du}{dx}=1 \\ ∴ du=dx \\ \begin{align*} \int (x\sqrt{x-1})dx &\to \int(u+1)(u^\frac{1}{2})du \\ &= \int(u^\frac{3}{2}+u^\frac{1}{2})du \\ &= \frac{2}{5}u^\frac{5}{2}+\frac{2}{3}u^\frac{3}{2}+C \\ &= \frac{2}{5}(x-1)^\frac{5}{2} + \frac{2}{3}(x-1)^\frac{3}{2} + C \end{align*} \]
Definite integration
To find a numerical value of the area under the curve in the bounded interval \([a,b]\), the definite integral can be taken. \[\int^b_a f(x)dx\]
\(a\) and \(b\) are known as the lower and upper limits of integration, respectively.
(Source;
Kognity)
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.
\[ \begin{align*} &\int^a_a f(x)dx&=&&0 \\ &\int^b_a c\cdot dx&=&&c(b-a) \\ &\int^a_b f(x)dx&=&&-\int^b_a f(x)dx \\ &\int^c_a f(x)dx&=&&\int^b_a f(x)dx + \int^c_b f(x)dx \end{align*} \]
The constant multiple and sum rules still apply.
Area between two curves
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.
Volumes of solids of revolution
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. \[ \begin{align*} V&=\lim_{x\to 0}\sum^b_{x=a}\pi y^2 dx \\ &=\pi\int^b_a y^2 dx \end{align*} \]
The area between two curves can also be rotated to form a solid, in which case its formula is: \[V=\pi\int^b_a \big[g(x)^2-f(x)^2\big]dx, g(x)>f(x)\]
Probability
!!! definition - \(\cap\) is the intersection sign and means “and”. - \(\cup\) is the union sign and means “or”. - \(\subset\) is the subset sign and indicates that the value on the left is a subset of the value on the right. - The sample space of an experiment is a list/set of all of the possible outcomes. - An event is a subset of a sample space that contains outcomes that meet a particular requirement.
Sets
A set is a collection of things represented with curly brackets that can be assigned to a variable.
!!! example \(A=\{0,1,2\}\) assigns the variable \(A\) to a collection of numbers \(0, 1, 2\).
The variable \(U\) is usually reserved for the universal set: a set that contains all of the elements under discussion for a particular situation.
Where both \(A\) and \(B\) are sets:
- \(A\cap B\) returns a new set with only objects that belong to both \(A\) and \(B\).
- \(A\cup B\) returns a new set with only objects that are inclusively in either \(A\) or \(B\).
- \(A\subset B\) is true only if all of the elements in \(A\) are also in \(B\).
- \(A'\) or \(A^c\) return the complement of a set: they return all elements in the universal set that are not in \(A\).
- \(n(A)\) returns the number of elements in set \(A\).
An empty/null set contains no objects and is represented either as \(\{\}\) or \(\emptyset\).
Two sets are disjoint or distinct if they have no common elements between them.
!!! warning Generally, unless specified otherwise, “between” should be inferred to mean “inclusively between”.
Probability rules
The probability of an event is represented by \(P(A)\), where \(A\) is the event. \[P(A)=\frac{n(A)}{n(U)}\]
As event \(A\) must be a subset of all possible outcomes \(U\), where \(1\) indicates that the event always happens and \(0\) the opposite: \[0\leq P(A)\leq 1\]
The complement of event \(A\) is the probability that it does not happen. It is written as \(A^c\), \(A'\), or \(\pu{not } A\). \[P(A')=1-P(A)\]
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: \(A\cup B=U\). \[P(A\cup B)=1\]
The principle of inclusion and exclusion forms a general rule for the union between two sets: \[P(A\cup B)=P(A)+P(B)-P(A\cap B)\]