internet intelligent tutors
Example 1 Find the derivative of \( \sin (2x-5) \) and use this result to deduce \( \displaystyle \int 10 \cos (2x-5) dx \). \( \begin{align} \displaystyle \frac{d}{dx} \sin (2x-5) &= \cos (2x-5) \times \frac{d}{dx} (2x-5) &\color{green}{\text{Chain Rule}} \\ &= \cos (2x-5) \times 2 \\ &= 2 \cos (2x-5) \\ \sin (2x-5) &= \int 2 […]
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:$$ \begin{align} \displaystyleV &= \int_{a}^{b}{\Big([f(x)]^2-[g(x)]^2\Big)}dx \\&= \int_{a}^{b}{\Big(y_{upper}^2-y_{lower}^2\Big)}dx\end{align} $$ Example 1 Find the volume of revolution generated by revolving the region between $y=x^2$ and […]
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:$$ \begin{align} \displaystyleV &= \lim_{h […]
Distances from Velocity Graphs Suppose a car travels at a constant positive velocity $80 \text{ km h}^{-1}$ for $2$ hours. $$ \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} $$We sketch the graph velocity against time, the graph is a horizontal line, and […]
If two functions $f(x)$ and $g(x)$ intersect at $x=1$ and $x=3$, and $f(x) \ge g(x)$ for all $1 \le x \le 3$, then the area of the shaded region between their points of intersection is given by:$$ \begin{align} \displaystyleA &= \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 bounded by the $x$-axis and $y=x^2-4x+3$.\( […]
$$ \large \begin{align} \displaystyleu &= f(x) \\du &= \dfrac{du}{dx} \times dx \\\end{align}$$ Converting $x$-values to corresponding $u$-values is required. Example 1 Find $\displaystyle 2\int_{0}^{1}{\sqrt{2x+1}}dx$. \( \begin{align} \displaystyle\text{Let } u &= 2x+1 \\\dfrac{du}{dx} &= 2 \\du &= 2dx \\u &= 2 \times 1 + 1 = 3 &x=1\\u &= 2 \times 0 + 1 = 1 […]
$$ \large \displaystyle \int_{n}^{m}{(ax+b)^k}dx = \dfrac{1}{a(k+1)}\Big[(ax+b)^{k+1}\Big]_{n}^{m}+c$$ 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} \\&= \bigg[\dfrac{(3x-2)^{-3}}{-9}\bigg]_{0}^{1} \\&= -\dfrac{1}{9}\big[(3x-2)^3\big]_{0}^{1} \\&= -\dfrac{1}{9}\big[(3 \times 1-2)^3-(3 \times 0-2)^3\big] \\&= -\dfrac{1}{9}(1 […]
$$ \large \begin{align} \displaystyle\int_{n}^{m}{\dfrac{1}{x}}dx &= \big[\log_e{x}\big]_{n}^{m} \\&= \log_{e}{m}-\log_{e}{n}\end{align} $$Generally,$$ \large \begin{align} \displaystyle\int_{n}^{m}{\dfrac{f'(x)}{f(x)}}dx &= \big[\log_e{f(x)}\big]_{n}^{m} \\&= \log_{e}{f(m)}-\log_{e}{f(n)}\end{align} $$ Example 1 Find $\displaystyle \int_{1}^{5}{\dfrac{2}{x}}dx$. \( \begin{align} \displaystyle\int_{1}^{5}{\dfrac{2}{x}}dx &= 2\int_{1}^{5}{\dfrac{1}{x}}dx \\&= 2\big[\log_{e}{x}\big]_{1}^{5} \\&= 2 \log_{e}{5}-2 \log_{e}{1} \\&= 2 \log_{e}{5}-2 \times 0 \\&= 2 \log_{e}{5} \end{align} \) Example 2 Find $\displaystyle \int_{2}^{8}{\dfrac{3x}{x^2+1}}dx$. \( \begin{align} \displaystyle\int_{2}^{8}{\dfrac{3x}{x^2+1}}dx &= \dfrac{3}{2} \int_{2}^{8}{\dfrac{2x}{x^2+1}}dx \\&= […]
$$ \large \begin{align} \displaystyle\int_{n}^{m}{e^{ax+b}}dx &= \dfrac{1}{a}\big[e^{ax+b}\big]_{n}^{m} \\&= \dfrac{1}{a}\big[e^{am+b}-e^{an+b}\big] \\\end{align} $$ Example 1 Find $\displaystyle \int_{2}^{4}{e^{2x-4}}dx$, leaving the answer in exact form. \( \begin{align} \displaystyle\int_{2}^{4}{e^{2x-4}}dx &= \dfrac{1}{2}\big[e^{2x-4}\big]_{2}^{4} \\&= \dfrac{1}{2}\big[e^{2 \times 4-4}-e^{2 \times 2-4}\big] \\&= \dfrac{e^4-e^{0}}{2} \\&= \dfrac{e^4-1}{2} \end{align} \) Example 2 Find $\displaystyle \int_{0}^{1}{(e^x+1)^2}dx$. \( \begin{align} \displaystyle\int_{0}^{1}{(e^{2x} + 2e^x + 1)}dx &= \big[\dfrac{1}{2}e^{2x} + 2e^x + […]
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 deducted from the fundamental theorem of calculus: Example 1 Prove $\displaystyle \int_{a}^{a}{f(x)}dx = 0$. \( \begin{align} \displaystyle\int_{a}^{a}{f(x)}dx &= \big[F(x)\big]_{a}^{a} \\&= F(a)-F(a) \\&= 0\end{align} \) Example […]