Area Between Two Functions

Area Between Two Functions

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 Find the area […]

Definite Integration of Power Functions

Definite Integration of Power Functions

$$\displaystyle \int_{n}^{m}{(ax+b)^k}dx = \dfrac{1}{a(k+1)}\Big[(ax+b)^{k+1}\Big]_{n}^{m}+c$$ Example 1 Find $\displaystyle \int_{0}^{1}{(2x+1)^5}dx$. \( \begin{align} \displaystyle \int{(2x+1)^5}dx &= \dfrac{(2x+1)^{5+1}}{2(5+1)} \\ &= \dfrac{1}{12}\big[(2x+1)^{6}\big]_{0}^{1} \\ &= \dfrac{1}{12}\big[(2 \times 1+1)^{6} – (2 \times 0+1)^{6}\big] \\ &= \dfrac{1}{12}(729 – 1) \\ &= \dfrac{728}{12} \\ &= \dfrac{182}{3} \end{align} \) Example 2 Find $\displaystyle \int_{0}^{1}{\dfrac{1}{(3x-2)^4}}dx$. \( \begin{align} \displaystyle \int_{0}^{1}{\dfrac{1}{(3x-2)^4}}dx &= \int_{0}^{1}{(3x-2)^{-4}}dx \\ &= \bigg[\dfrac{(3x-2)^{-4+1}}{3(-4+1)}\bigg]_{0}^{1} \\ […]

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 $\displaystyle \int_{1}^{5}{\dfrac{2}{x}}dx$. \( \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} – 2 \times […]

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. \( \begin{align} \displaystyle \int_{2}^{4}{e^{2x-4}}dx &= \dfrac{1}{2}\big[e^{2x-4}\big]_{2}^{4} \\ &= \dfrac{1}{2}\big[e^{2 \times 4 – 4} – e^{2 \times 2 – 4}\big] \\ &= \dfrac{e^4 – e^{0}}{2} \\ &= \dfrac{e^4 – 1}{2} \\ \end{align} \) […]

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) […]

Integration of Power Functions

Integration of Power Functions

$$\displaystyle \int{(ax+b)^n}dx = \dfrac{(ax+b)^{n+1}}{a(n+1)}+c$$ Example 1 Find $\displaystyle \int{(2x+1)^5}dx$. \( \begin{align} \displaystyle \int{(2x+1)^5}dx &= \dfrac{(2x+1)^{5+1}}{2(5+1)} +c \\ &= \dfrac{(2x+1)^{6}}{12} +c \\ \end{align} \) Example 2 Find $\displaystyle \int{\dfrac{1}{(3x-2)^4}}dx$. \( \begin{align} \displaystyle \int{\dfrac{1}{(3x-2)^4}}dx &= \int{(3x-2)^{-4}}dx \\ &= \dfrac{(3x-2)^{-4+1}}{3(-4+1)} +c\\ &= \dfrac{(3x-2)^{-3}}{-9} +c\\ &= -\dfrac{1}{9(3x-2)^3} +c\\ \end{align} \) Example 3 Find $\displaystyle \int{\sqrt{4x+3}}dx$. \( \begin{align} \displaystyle \int{\sqrt{4x+3}}dx […]