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Volumes for Two Functions

Volumes for Two Functions

Home 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 volume of [...]
Area Between Two Functions

Area Between Two Functions

Home 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 \( \begin{align} \displaystyle [...]
Definite Integration of Power Functions

Definite Integration of Power Functions

Home $$\displaystyle \int_{n}^{m}{(ax+b)^n}dx = \dfrac{1}{a(n+1)}\big[(ax+b)^{n+1}\big]_{n}^{m}+c$$ 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 \int_{0}^{1}{\dfrac{1}{(3x-2)^4}}dx &= [...]
Definite Integral of Rational Functions

Definite Integral of Rational Functions

Home $$ \begin{align} \displaystyle \int_{n}^{m}{\dfrac{f'(x)}{f(x)}}dx &= \big[\log_e {f(x)}\big]_{n}^{m} \\ &= \log_e{f(m)} - \log_e{f(n)} \end{align} $$ 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} \displaystyle \int_{1}^{3}{\dfrac{2}{2x-1}}dx [...]
Definite Integral of Exponential Functions

Definite Integral of Exponential Functions

Home $$ \begin{align} \displaystyle \int_{n}^{m}{e^{ax+b}}dx &= \dfrac{1}{a}\big[e^{ax+b}\big]_{n}^{m} \\ &= \dfrac{1}{a}\big[e^{am+b}-e^{an+b}\big] \\ \end{align} $$ 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 - 1}{2} [...]
Definite Integrals

Definite Integrals

Home 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: $\displaystyle \int_{a}^{a}{f(x)}dx = 0$ $\displaystyle \int_{b}^{a}{f(x)}dx = -\int_{a}^{b}{f(x)}dx$ $\displaystyle \int_{b}^{a}{f(x)}dx + \int_{c}^{b}{f(x)}dx = \int_{c}^{a}{f(x)}dx$ $\displaystyle [...]
Trigonometric Integration by Substitution

Trigonometric Integration by Substitution

Home 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} \\ -du &= [...]
Integration of Power Functions

Integration of Power Functions

Home $$\displaystyle \int{(ax+b)^n}dx = \dfrac{(ax+b)^{n+1}}{a(n+1)}+c$$ 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 Find $\displaystyle [...]
Integration of Rational Functions

Integration of Rational Functions

Home 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 &= \dfrac{1}{3}\int{\dfrac{1}{x}}dx [...]
Integration of Trigonometric Functions

Integration of Trigonometric Functions

Home 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 \\ \therefore [...]
Integration of Exponential Functions

Integration of Exponential Functions

Home 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} \times a [...]