Volumes for Two Functions

Volumes for Two Functions

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 revolution [...]
Area Between Two Functions

Area Between Two Functions

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: $$ \begin{align} \displaystyle A &= \int_{1}^{3}{f(x)}dx - \int_{1}^{3}{g(x)}dx \\ &= \int_{1}^{3}{\Big[f(x)-g(x)\Big]}dx \end{align} $$ Example 1 [...]
Definite Integration of Power Functions

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

Definite Integral of Rational Functions

$$ \begin{align} \displaystyle \int_{n}^{m}{\dfrac{1}{x}}dx &= \big[\log_e{x}\big]_{n}^{m} \\ &= \log_{e}{m} - \log_{e}{n} \end{align} $$ Generally, $$ \begin{align} \displaystyle \int_{n}^{m}{\dfrac{f'(x)}{f(x)}}dx &= \big[\log_e{f(x)}\big]_{n}^{m} \\ &= \log_{e}{f(m)} - \log_{e}{f(n)} \end{align} $$ Example 1 Find $\displaystyle \int_{1}^{5}{\dfrac{2}{x}}dx$. Show Solution \( \begin{align} \displaystyle \int_{1}^{5}{\dfrac{2}{x}}dx &= 2\int_{1}^{5}{\dfrac{1}{x}}dx \\ &= 2\big[\log_{e}{x}\big]_{1}^{5} \\ &= 2 \log_{e}{5} - 2 \log_{e}{1} \\ &= 2 \log_{e}{5} - [...]
Definite Integral of Exponential Functions

Definite Integral of Exponential Functions

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

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 \int_{b}^{a}{\big[f(x) [...]