Integral Calculus

calculus year12 advanced

Integral calculus | NSW Curriculum Website

• MAV-12-05 solves problems involving indefinite and definite integrals

📖 Prior Knowledge

Content	Prior knowledge	Used for
Area	- area of a trapezium	- trapezoidal rule
Non-Linear Relationships C	- points of intersection	- finding area bounded between two curves
Differential Calculus	- differential calculus	- integral calculus

Primitive functions

• Define a primitive of a function $f(x)$ as a function $F(x)$ whose derivative $F^{\prime }(x)=f(x)$ and recognise the process of finding the primitive as the reverse of differentiation
• Recognise that a function whose derivative is everywhere zero is a constant function
• Prove by differentiation that a primitive of $f(x)=x^n$ is $F(x)=\frac{x^{n+1}}{n+1}$, for all real $n\ne -1$
• Prove by differentiation that if $F(x)$ and $G(x)$ are primitives of $f(x)$ and $g(x)$, and $k$ is a constant, then $F(x)+G(x)$ is a primitive of $f(x)+g(x)$, and $kF(x)$ is a primitive of $kf(x)$
• Recognise that primitives of a function $f(x)$ are not unique, and that any two primitives of $f(x)$ differ by a constant, so that if $F(x)$ is a primitive of $f(x)$, the general primitive of $f(x)$ is $F(x)+C$, for some constant $C$
• Determine the primitive of a given function $f(x)$, where $f(x)$ is a sum of functions of the form $k{x}^n$ for all real $n\ne -1$
• Determine the primitive function for functions of the form $f(x)=(ax+b{)}^n$, for all real $n\ne -1$, where $a$ and $b$ are constants
• Use algebraic manipulation to express given functions in forms suitable for determining primitive functions
• Determine $f(x)$, given $f^{\prime }(x)$ and an initial condition $f(a)=b$ where $a$ and $b$ are constants

The definite integral

• Examine for a function $f(x)$, which indicates the rate of change of a quantity, the meaning of $\sum _a^{b}f(x)\Delta x$, where the interval $a\le x\le b$ is divided into subintervals of length $\Delta x$, and describe $\sum _a^{b}f(x)\Delta x$ as an estimate of the total change in that quantity over the interval $a\le x\le b$
• Consider the definite integral as ${\int }_a^{b}f(x)dx=\underset{\Delta x\to 0}{\lim }\sum _a^{b}f(x)\Delta x$, noting that this implies that the result of a definite integral will be negative when $f(x)\le 0$ throughout the interval $a\le x\le b$
• Define informally that a function is continuous on the interval $a\le x\le b$ if it can be drawn between the two endpoints of the interval without taking the pen off the paper
• Graph the region between the continuous function $y=f(x)$ and the $x$-axis, where $f(x)\ge 0$ on the interval $a\le x\le b$
• Use a graphing application to compare different methods of approximating the area, $A$, of the region between the continuous function $y=f(x)$ and the $x$-axis, where $f(x)\ge 0$ on the interval $a\le x\le b$, by summing the areas of trapezia or rectangles each of width $\Delta x=\frac{b-a}{n}$ and approximate height $f(x)$ for any $x$ lying in its base, and observe the effect on the precision of the approximation of $A$ as the number $n$ of subintervals of $a\le x\le b$ increases, that is as $\Delta x\to 0$
• Evaluate the definite integral ${\int }_a^{b}f(x)dx$ by calculating areas using geometrical formulas, where the shape of $f(x)$ allows such calculations, in cases where $f(x)\ge 0$ throughout $a\le x\le b$, $f(x)\le 0$ throughout $a\le x\le b$ or where $f(x)$ changes sign in the interval $a\le x\le b$

The Fundamental Theorem of Calculus

• Consider the function defined by $A(x)={\int }_a^{x}f(t)dt$ and use a graphing application to recognise that $A(x)$ is a primitive of $f(x)$
• Recognise the Fundamental Theorem of Calculus as ${\int }_a^{b}f(x)dx=[F(x){]}_a^b=F(b)-F(a)$ for a continuous function $f$ on the interval $a\le x\le b$ where $F(x)$ is any primitive of $f(x)$

Indefinite integrals

• Use the notation $\int f(x)dx$ for the general primitive of $f(x)$, called the indefinite integral of $f(x)$, so that $\int f(x)dx=F(x)+C$, for some constant $C$, where $F(x)$ is any primitive of $f(x)$
• Recognise integration as the process of finding the indefinite integral of a function
• Use the formula $\int x^{n}dx=\frac{1}{n+1}{x}^{n+1}+C$ for real $n\ne -1$
• Use the identities $\int (f(x)+g(x))dx=\int f(x)dx+\int g(x)dx$ and $\int kf(x)dx=k\int f(x)dx$ for primitives
• Prove by differentiation, and apply $\int u^n\frac{du}{dx}dx=\frac{1}{n+1}{u}^{n+1}+C$, where $u$ is a function of $x$, or $\int f^{\prime }(x)[f(x){]}^{n}dx=\frac{1}{n+1}[f(x){]}^{n+1}+C$, for real $n\ne -1$

Integration with exponential functions

• Establish and use the formula $\int e^{x}dx=e^x+C$
• Establish and use the formula $\int e^{ax+b}dx=\frac{1}{a}{e}^{ax+b}+C$, where $a$ and $b$ are constants and $a\ne 0$
• Establish and use the formula $\int a^{x}dx=\frac{a^x}{\ln a}+C$, where $a$ is a constant and $a>0$
• Establish and use $\int e^u\frac{du}{dx}dx=e^u+C$, where $u$ is a function of $x$, or $\int f^{\prime }(x)e^{f(x)}dx=e^{f(x)}+C$
• Find primitives of functions involving exponential functions

Integration with logarithmic functions

• Derive and use the formula $\int \frac{1}{x}dx=\ln |x|+C$ where $x\ne 0$
• Establish and use the formula $\int \frac{1}{ax+b}dx=\frac{1}{a}\ln |ax+b|+C$, where $a$ and $b$ are constants and $a\ne 0$
• Establish and use $\int \frac{u^{\prime }}{u}dx=\ln |u|+C$, where $u$ is a function of $x$, or $\int \frac{f^{\prime }(x)}{f(x)}dx=\ln |f(x)|+C$, on a domain where $f(x)\ne 0$

Integration with trigonometric functions

• Establish and use the formulas $\int \sin xdx=-\cos x+C$, $\int \cos xdx=\sin x+C$ and $\int {\sec }^{2}xdx=\tan x+C$
• Establish and use indefinite integrals of the form $\int f(ax+b)dx$, where $a$ and $b$ are constants and $a\ne 0$, and $f(x)=\sin x$, $f(x)=\cos x$ and $f(x)={\sec }^{2}x$
• Determine indefinite integrals of the form $\int f^{\prime }(x)\sin f(x)dx$, $\int f^{\prime }(x)\cos f(x)dx$ and $\int f^{\prime }(x){\sec }^{2}f(x)dx$

Areas and the definite integral

• Apply ${\int }_a^{b}f(x)dx=F(b)-F(a)$, where $F(x)$ is a primitive of $f(x)$, to calculate definite integrals and solve related theoretical problems involving functions within the scope of the Mathematics Advanced course
• Describe, in the case where $f(x)\ge 0$ for all values of $x$ in the interval $a\le x\le b$, the area bounded by the graph of the continuous function $y=f(x)$, the $x$-axis and the lines $x=a$ and $x=b$, as ${\int }_a^{b}f(x)dx$
• Recognise, in the case where $f(x)\le 0$ for all values of $x$ in the interval $a\le x\le b$, the area bounded by the graph of the continuous function $y=f(x)$, the $x$-axis and the lines $x=a$ and $x=b$, as $|{\int }_a^{b}f(x)dx|$ or $-{\int }_a^{b}f(x)dx$
• Conclude, for a continuous function $y=f(x)$ on the interval $a\le x\le b$, that ${\int }_a^{b}f(x)dx=$ (area of regions between curve and $x$-axis lying above the $x$-axis) - (area of regions between curve and the $x$-axis lying below the $x$-axis)
• Use definite integrals to solve problems involving the areas of regions bounded by the graph of the continuous function $y=f(x)$, the $x$-axis and the lines $x=a$ and $x=b$, in cases where $f(x)\ge 0$ throughout $a\le x\le b$, $f(x)\le 0$ throughout $a\le x\le b$ or where $f(x)$ changes sign in the interval $a\le x\le b$, with or without the graph provided
• Use definite integrals to solve problems involving the areas of regions bounded by the graph of the continuous function $y=f(x)$, the $y$-axis and the lines $y=a$ and $y=b$, in cases where $x\ge 0$ throughout $a\le y\le b$, $x\le 0$ throughout $a\le y\le b$ or where $x$ changes sign in the interval $a\le y\le b$ with or without the graph provided
• Use the fact that the graphs of $y=a^x$ and $y={\log }_{a}x$ are reflections of each other in the line $y=x$ to solve problems involving areas between the $x$-axis or $y$-axis and a curve involving either an exponential or logarithmic function
• Recognise and use the result, where $f(x)$ is continuous on the interval $a\le x\le c$, ${\int }_a^{b}f(x)dx+{\int }_b^{c}f(x)dx={\int }_a^{c}f(x)dx$ for all $c$ such that $a\le b\le c$
• Define and use the result ${\int }_a^{b}f(x)dx=-{\int }_b^{a}f(x)dx$, where $f(x)$ is continuous on the interval $a\le x\le b$
• Recognise and use symmetry, particularly odd and even functions, to simplify and solve integration problems
• Use the Trapezoidal rule to approximate integrals
• Use an online computational application to evaluate definite and indefinite integrals involving functions within and beyond the scope of the Mathematics Advanced course
• Model and solve practical problems involving integrals and areas of regions bounded by a curve and the $x$-axis, or by a curve and the $y$-axis, involving functions within the scope of the Mathematics Advanced course