Applications of the definite integral

1. The area under a graph ${{\Gamma }_{f}}$, $f:[a,b]\to {{\mathbb{R}}_{+}}, f\text{ continu }\!\!\breve{\mathrm{a}}\!\!$

$\displaystyle {{\Gamma }_{f}}=\int\limits_{a}^{b}{\left| f(x) \right|} dx$

The area between the graphs of two continuous functions $f,g:[a,b]\to \mathbb{R}$ is:

$\displaystyle {{\Gamma }_{f,g}}=\sum\limits_{k=1}^{n}{\left[ f\left( a+k\frac{b-a}{n} \right)-g\left( a+k\frac{b-a}{n} \right) \right]}\cdot \frac{b-a}{n}$

$\displaystyle {{\Gamma }_{f,g}}=\int\limits_{a}^{b}{\left| f(x)-g(x) \right|} dx$

The region between the two graphs is the set of points:

$\Gamma =\{(x,y)\in {{\mathbb{R}}^{2}}|a\le x\le b\text{ i }g(x)\le y\le f(x)\}$

The area of the region lying between the graphs of the functions

$f(x)=x$ and $g(x)={{x}^{2}}$

The area of the region lying between the graphs of the functions

$f(x)={{x}^{2}}$ and $g(x)={{x}^{3}}$

|:--:| | $\sin (x)$ | | The region enclosed between the graphs of the functions $\sin (x)$ and $co\operatorname{s}(x)$ |

|:--:|

| | The region enclosed between the graphs of the functions and $\sqrt[3]{x}$ |

2. Arc length of a graph

Let $f:[a,b]\to \mathbb{R}$ be a differentiable function with continuous derivative. The length of the graph of $f$ is:

$\displaystyle l=\int_{a}^{b}{\sqrt{1+{{({f}'(x))}^{2}}}}dx$

3. Area of a surface of revolution

Let $f:[a,b]\to \mathbb{R}$ be a continuous function. The set of points in space:

$S=\{(x,y,z)\in {{\mathbb{R}}^{3}}|\sqrt{{{y}^{2}}+{{z}^{2}}}=f(x),x\in [a,b]\}$

is called the surface of revolution determined by the function $f$ about the $Ox$ axis.

Let $f:[a,b]\to \mathbb{R}$ be a differentiable function with continuous derivative. The surface of revolution determined by $f$ has area:

$\displaystyle A(f)=2\pi \int_{a}^{b}{f(x)\cdot \sqrt{1+{{({f}'(x))}^{2}}}}dx$

The surface of revolution of the function $\sqrt{x}$ on the interval $[0,9]$

The surface of revolution of the function cos(x) on the interval $[0,2\pi ]$

The surface of revolution of the function sin(x) on the interval $[0,2\pi ]$

The surface of revolution of the function $x\cdot \sin (x)$ on the interval $[0,2\pi ]$

The surface of revolution of the function ${{x}^{2}}$ on the interval $[-2,4]$

The surface of revolution of the function ${{x}^{3}}-{{x}^{2}}$ on the interval $[-1,2]$

4. Volume of a solid of revolution

Let $f:[a,b]\to \mathbb{R}$ be a continuous function. The set of points in space:

$V=\{(x,y,z)\in {{\mathbb{R}}^{3}}|\sqrt{{{y}^{2}}+{{z}^{2}}}\le \left| f(x) \right|,x\in [a,b]\}$

is called the solid of revolution determined by the function $f$ about the $Ox$ axis.

Let $f:[a,b]\to \mathbb{R}$ be a differentiable function with continuous derivative. The solid of revolution determined by $f$ has volume:

$\displaystyle V(f)=\pi \int_{a}^{b}{{{f}^{2}}(x)}dx$

5. Centre of mass

Let $E$ be a plane lamina bounded by the continuous functions $f,g:[a,b]\to \mathbb{R}.$ The centre of mass of $E$ is the point with coordinates:

$\displaystyle {{x}_{G}}=\tfrac{\int\limits_{a}^{b}{x(g(x)-f(x))dx}}{\int\limits_{a}^{b}{(g(x)-f(x))dx}}$, $\displaystyle {{y}_{G}}=\tfrac{\frac{1}{2}\cdot \int\limits_{a}^{b}{({{g}^{2}}(x)-{{f}^{2}}(x))dx}}{\int\limits_{a}^{b}{(g(x)-f(x))dx}}$