Applications of the definite integral

1. The area under a graph ,

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

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

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

$f(x)=x$ and

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

and

|:--:| | $sin(x)$ | | The region enclosed between the graphs of the functions $sin(x)$ and $cos(x)$ |

|:--:|

| | The region enclosed between the graphs of the functions and |

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:

3. Area of a surface of revolution

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

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:

The surface of revolution of the function 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 on the interval $[-2,4]$

The surface of revolution of the function 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:

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:

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:

,