Chapter Four - Integration

4A - Areas and the definite integral

all area formulae and calculations of area are based on two principles:

1. area of a rectangle = length x breadth

2. when a region is dissected, the area is unchanged.
   

The definite integral:$\int f\left(x\right)$ dx, is defined as the area of the region between the curve and x-axis, from x=a, to x=b

• the function $f\left(x\right)$ is called the integrand, an the values x = a and x = b are called the lower and upper limits of the integral.

Area formulas:

Triangle: $\frac12b\times h$

Trapezium: $\frac12\left(a+b\right)h$

Circle: $\pi r^2$

4B - The fundamental theorem of calculus

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a function $F(x)$ is called a primitive or anti-derivative of a function $f(x)$ if its derivative is f(x): $F(x)$ is primitive or $f(x)$ if $F’(x) = f(x)$ .

To find the general primitive of a power dy/dx = $x^{n}$ , then y = $\frac{x^{n+1}}{n+1}+C$ for some constant C.

• increase the index by 1 and divide by the new index.

let $f\left(x\right)$ be a function that is continuous in a closed interval [a, b]. then

$\,\int_{a}^{b}f\left(x\right)\!\,dx=F\left(b\right)-F\left(a\right)$ where F(x) is any primitive of f(x).

4C - The definite integral and its properties

integrating functions with negative values

when a function has negative values, its graph is below the x-axis, so the ‘heights’