#Calculus #Year12 #Advanced >[!info]- [Integral calculus | NSW Curriculum Website](https://curriculum.nsw.edu.au/learning-areas/mathematics/mathematics-advanced-11-12-2024/content/n12/fa41f9f88f) >- 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 $F'(x)=f(x)$ 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 $n \neq -1$ - 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 $kx^n$ for all real $n \neq -1$ - Determine the primitive function for functions of the form $f(x)=(ax+b)^n$, for all real $n \neq -1$, where $a$ and $b$ are constants - Use algebraic manipulation to express given functions in forms suitable for determining primitive functions - Determine $f(x)$, given $f'(x)$ 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 $a \leq x \leq b$ 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 $a \leq x \leq b$ - Consider the definite integral as $\int_{a}^{b}{f(x)}dx=\lim_{\Delta x \to 0}{\sum_{a}^{b}f(x)\Delta x}$, noting that this implies that the result of a definite integral will be negative when $f(x) \leq 0$ throughout the interval $a \leq x \leq b$ - Define informally that a function is continuous on the interval $a \leq x \leq b$ 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 $f(x) \geq 0$ on the interval $a \leq x \leq b$ - 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 $f(x) \geq 0$ on the interval $a \leq x \leq b$, 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 $a \leq x \leq b$ increases, that is as $\Delta x \to 0$ - Evaluate the definite integral $\int_a^bf(x)dx$ by calculating areas using geometrical formulas, where the shape of $f(x)$ allows such calculations, in cases where $f(x) \geq 0$ throughout $a \leq x \leq b$, $f(x) \leq 0$ throughout $a \leq x \leq b$ or where $f(x)$ changes sign in the interval $a \leq x \leq b$ ## 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 $a \leq x \leq b$ 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 $\int x^ndx=\frac{1}{n+1}x^{n+1}+C$ for real $n \neq -1$ - 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 $\int f'(x)[f(x)]^ndx=\frac{1}{n+1}[f(x)]^{n+1}+C$, for real $n \neq -1$ ## Integration with exponential functions - Establish and use the formula $\int e^xdx=e^x+C$ - Establish and use the formula $\int e^{ax+b}dx=\frac{1}{a}e^{ax+b}+C$, where $a$ and $b$ are constants and $a \neq 0$ - Establish and use the formula $\int a^xdx=\frac{a^x}{\ln a}+C$, 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 $\int f'(x)e^{f(x)}dx=e^{f(x)}+C$ - 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 $x \neq 0$ - 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 $a \neq 0$ - Establish and use $\int\frac{u'}{u}dx=\ln{|u|}+C$, where $u$ is a function of $x$, or $\int\frac{f'(x)}{f(x)}dx=\ln{|f(x)|}+C$, on a domain where $f(x) \neq 0$ ## Integration with trigonometric functions - Establish and use the formulas $\int\sin{x}dx=-\cos{x}+C$, $\int\cos{x}dx=\sin{x}+C$ and $\int\sec^2{x}dx=\tan{x}+C$ - Establish and use indefinite integrals of the form $\int f(ax+b)dx$, where $a$ and $b$ are constants and $a \neq 0$, and $f(x)=\sin{x}$, $f(x)=\cos{x}$ and $f(x)=\sec^2{x}$ - Determine indefinite integrals of the form $\int f'(x)\sin{f(x)}dx$, $\int f'(x)\cos{f(x)}dx$ and $\int f'(x)\sec^2{f(x)}dx$ ## 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 $f(x) \geq 0$ for all values of $x$ in the interval $a \leq x \leq b$, 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 $f(x) \leq 0$ for all values of $x$ in the interval $a \leq x \leq b$, 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 $a \leq x \leq b$, 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 $f(x) \geq 0$ throughout $a \leq x \leq b$, $f(x) \leq 0$ throughout $a \leq x \leq b$ or where $f(x)$ changes sign in the interval $a \leq x \leq b$, 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 $x \geq 0$ throughout $a \leq y \leq b$, $x \leq 0$ throughout $a \leq y \leq b$ or where $x$ changes sign in the interval $a \leq y \leq b$ 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 $a \leq x \leq c$, $\int_{a}^{b}f(x)dx+\int_{b}^{c}f(x)dx=\int_{a}^{c}f(x)dx$ for all $c$ such that $a \leq b \leq c$ - 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 $a \leq x \leq b$ - 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