 Volumes for Two Functions

Home > Integration If the region bounded by the upper function $y_{upper}=f(x)$ and the lower funciton $y_{lower}=g(x)$, and the lines $x=a$ and $x=b$ is revolved about the $x$-axis, then its volume of revolution is given by: \begin{align} \displaystyle V &= \int_{a}^{b}{\Big([f(x)]^2 - [g(x)]^2\Big)}dx \\ &= \int_{a}^{b}{\Big(y_{upper}^2 - y_{lower}^2\Big)}dx \end{align} Example 1 Find the [...] Volumes using Integration

Home > Integration Volume of Revolution We can use integration to find volumes of revolution between $x=a$ and $x=b$. When the region enclosed by $y=f(x)$, the $x$-axis, and the vertical lines $x=a$ and $x=b$ is revolved through $2 \pi$ or $360^{\circ}$about the $x$-axis to generate a solid, the volume of the solid is given by: [...] Kinematics using Integration

Home > Integration Distances from Velocity Graphs Suppose a car travels at a constant positive velocity $80 \text{ km h}^{-1}$ for $2$ hours. \begin{align} \displaystyle \text{distance travelled} &= \text{speed} \times \text{time} \\ &= 80 \text{ km h}^{-1} \times 2 \text{ h} \\ &= 160 \text{ km} \end{align} We we sketch the graph velocity [...] Home > Integration If two functions $f(x)$ and $g(x)$ intersect at $x=1$ and $x=3$, and $f(x) \ge g(x)$ for all $1 \le x \le 3$, then the area of the shaded region between their points of intersection is given by: $$A=\int_{1}^{3}{\big[f(x)-g(x)\big]}dx$$ Example 1 Find the area bounded by the $x$-axis and $y=x^2-4x+3$. Show Solution [...] Definite Integration by Substitution Home > Integration  \begin{align} \displaystyle u &= f(x) \\ du &= \dfrac{du}{dx} \times dx \\ \end{align} Converting x-values to corresponding u-values are required. Example 1 Find \displaystyle 2\int_{0}^{1}{\sqrt{2x+1}}dx. Show Solution \( \begin{align} \displaystyle \text{Let } u &= 2x+1 \\ \dfrac{du}{dx} &= 2 \\ du &= 2dx \\ u &= 2 \times 1 + 1 [...] Definite Integration of Power Functions Home > Integration \displaystyle \int_{n}^{m}{(ax+b)^k}dx = \dfrac{1}{a(k+1)}\Big[(ax+b)^{k+1}\Big]_{n}^{m}+c Example 1 Find \displaystyle \int_{0}^{1}{(2x+1)^5}dx. Show Solution \( \begin{align} \displaystyle \int{(2x+1)^5}dx &= \dfrac{(2x+1)^{5+1}}{2(5+1)} \\ &= \dfrac{1}{12}\big[(2x+1)^{6}\big]_{0}^{1} \\ &= \dfrac{1}{12}\big[(2 \times 1+1)^{6} - (2 \times 0+1)^{6}\big] \\ &= \dfrac{1}{12}(729 - 1) \\ &= \dfrac{728}{12} \\ &= \dfrac{182}{3} \end{align} Example 2 Find $\displaystyle \int_{0}^{1}{\dfrac{1}{(3x-2)^4}}dx$. Show Solution \begin{align} \displaystyle [...] Definite Integral of Rational Functions Home > Integration  \begin{align} \displaystyle \int_{n}^{m}{\dfrac{f'(x)}{f(x)}}dx &= \big[\log_e {f(x)}\big]_{n}^{m} \\ &= \log_e{f(m)} - \log_e{f(n)} \end{align}  Example 1 Find \displaystyle \int_{1}^{2}{\dfrac{1}{x}}dx. Show Solution \( \begin{align} \displaystyle \int_{1}^{2}{\dfrac{1}{x}}dx &= \big[\log_e{x}\big]_{1}^{2} \\ &= \log_e{2} - \log_e{1} \\ &= \log_e{2} - 0 \\ &= \log_e{2} \\ \end{align} Example 2 Find $\displaystyle \int_{1}^{3}{\dfrac{2}{2x-1}}dx$. Show Solution \begin{align} [...] Definite Integral of Exponential Functions Home > Integration  \begin{align} \displaystyle \int_{n}^{m}{e^{ax+b}}dx &= \dfrac{1}{a}\big[e^{ax+b}\big]_{n}^{m} \\ &= \dfrac{1}{a}\big[e^{am+b}-e^{an+b}\big] \\ \end{align}  Example 1 Find \displaystyle \int_{2}^{4}{e^{2x-4}}dx, leaving the answer in exact form. Show Solution \( \begin{align} \displaystyle \int_{2}^{4}{e^{2x-4}}dx &= \dfrac{1}{2}\big[e^{2x-4}\big]_{2}^{4} \\ &= \dfrac{1}{2}\big[e^{2 \times 4 - 4} - e^{2 \times 2 - 4}\big] \\ &= \dfrac{e^4 - e^{0}}{2} \\ &= \dfrac{e^4 [...] Definite Integrals Home > Integration The Fundamental Theorem of Calculus For a continuous function f(x) with antiderivative F(x), \displaystyle \int_{a}^{b}{f(x)}dx = F(b) - F(a) Properties of Definite Integrals The following properties of definite integrals can all be deductefd from the fundamental theorem of calculus: \displaystyle \int_{a}^{a}{f(x)}dx = 0 \displaystyle \int_{b}^{a}{f(x)}dx = -\int_{a}^{b}{f(x)}dx \displaystyle \int_{b}^{a}{f(x)}dx + \int_{c}^{b}{f(x)}dx = [...] Area Under a Curve using Integration Home > Integration If f(x) is positive and continuous on the interval 1 \le x \le 4, then the area bounded by y=f(x), the x-axis, and the vertical lines x=1 and x=4 is given by: \displaystyle A=\int_{1}^{4}{f(x)}dx Example 1 Find the area of the region bounded by y=2x, x-axis, x=1 and x=5. Show Solution \( [...] Trigonometric Integration by Substitution Home > Integration Substitution of \sin{x}  \begin{align} \displaystyle \text{ Let} u &= \sin{x} \\ \dfrac{du}{dx} &= \cos{x} \\ du &= \cos{x}du \\ \int{\sin^n{x}\cos{x}}dx &= \int{u^n}du \\ &= \dfrac{u^{n+1}}{n+1} + c \\ &= \dfrac{\sin^{n+1}{x}}{n+1} + c \\ \end{align}  Substitution of \cos{x}  \begin{align} \displaystyle \text{ Let} u &= \cos{x} \\ \dfrac{du}{dx} &= -\sin{x} \\ [...] Integration of Power Functions Home > Integration \displaystyle \int{(ax+b)^n}dx = \dfrac{(ax+b)^{n+1}}{a(n+1)}+c Example 1 Find \displaystyle \int{(2x+1)^5}dx. Show Solution \( \begin{align} \displaystyle \int{(2x+1)^5}dx &= \dfrac{(2x+1)^{5+1}}{2(5+1)} +c \\ &= \dfrac{(2x+1)^{6}}{12} +c \\ \end{align} Example 2 Find $\displaystyle \int{\dfrac{1}{(3x-2)^4}}dx$. Show Solution \begin{align} \displaystyle \int{\dfrac{1}{(3x-2)^4}}dx &= \int{(3x-2)^{-4}}dx \\ &= \dfrac{(3x-2)^{-4+1}}{3(-4+1)} +c\\ &= \dfrac{(3x-2)^{-3}}{-9} +c\\ &= -\dfrac{1}{9(3x-2)^3} +c\\ \end{align} Example 3 [...] Home > Integration We can find the integral constant $c$, given a particular value of the function. Example 1 Find $f(x)$ given that $f'(x) = 3x^2+4x-5$ and $f(1)=3$. Show Solution \begin{align} \displaystyle f(x) &= \int{(3x^2+4x-5)}dx \\ &= \dfrac{3x^{2+1}}{2+1} + \dfrac{4x^{1+1}}{1+1} - 5x +c \\ &= x^3 + 2x^2 - 5x + c \\ f(1) [...] Integration using Double Angle Formula Home > Integration  \begin{align} \displaystyle \cos{2x} &= 2\cos^2{x} - 1 \\ &= 1-2 \sin^2 {x} \\ &= \cos^2{x} - \sin^2{x} \\ \sin{2x} &= 2 \sin{x} \cos{x} \\ \end{align}  Example 1 Find \displaystyle \int{\sin^2{x}}dx. Show Solution \( \begin{align} \displaystyle \cos{2x} &= 1 - 2 \sin^2{x} \\ 2 \sin^2{x} &= 1 - \cos{2x} \\ \sin^2{x} [...] Integration of Rational Functions Home > Integration Integration of \displaystyle \dfrac{1}{x}  \begin{align} \displaystyle \dfrac{d}{dx}\log_ex &= \dfrac{1}{x} \\ \log_ex &= \int{\dfrac{1}{x}}dx \\ \therefore \int{\dfrac{1}{x}}dx &= \log_ex +c\\ \end{align}  Example 1 Find \displaystyle \int{\dfrac{2}{x}}dx. Show Solution \( \begin{align} \displaystyle \int{\dfrac{2}{x}}dx &= 2\int{\dfrac{1}{x}}dx \\ &= 2\log_ex +c\\ \end{align} Example 2 Find $\displaystyle \int{\dfrac{1}{3x}}dx$. Show Solution \( \begin{align} \displaystyle \int{\dfrac{1}{3x}}dx [...] Home > Integration Integration of $\cos{x}$ \begin{align} \displaystyle \dfrac{d}{dx}\sin{x} &= \cos{x} \\ \sin{x} &= \int{\cos{x}}dx \\ \therefore \int{\cos{x}}dx &= \sin{x} +c \\ \end{align} Integration of $\cos{(ax+b)}$ \begin{align} \displaystyle \dfrac{d}{dx}\sin{(ax+b)} &= \cos{(ax+b)} \times \dfrac{d}{dx}(ax+b) \\ &= \cos{(ax+b)} \times a \\ &= a\cos{(ax+b)} \\ \sin{(ax+b)} &= \int{a\cos{(ax+b)}}dx \\ &= a\int{\cos{(ax+b)}}dx \\ \dfrac{1}{a}\sin{(ax+b)} &= \int{\cos{(ax+b)}}dx [...] Integration of Exponential Functions Home > Integration The base formula of integrating exponential function is obtained from deriving e^x. \begin{align} \displaystyle \dfrac{d}{dx}e^x &= e^x \\ e^x &= \int{e^x}dx \\ \therefore \int{e^x}dx &= e^x +c \\ \end{align} $$This base formula is extended to the following general formula.$$ \begin{align} \displaystyle \dfrac{d}{dx}e^{ax+b} &= e^{ax+b} \times \dfrac{d}{dx}(ax+b) \\ &= e^{ax+b} [...] Calculation of Areas under Curves

Home > Integration Consider the function $f(x)=x^2+2$. We wish to estimate the green area enclosed by $y=f(x)$, the $x$-axis, and the vertical lines $x=1$ and $x=4$. Suppose we divide the $x$-interval into three strips of width 1 unit. Upper Rectangles The diagram below shows upper rectangles, which are rectangles with top edges at the maximum [...]