We can find the area between a function f and the x-axis by either evaluating the sum of rectangular strips with a reactangular area, and finding an approximation. Or, more precisely find the limit of all strips n tend to infinity.
TODO - see Riemann integral
fundamental theorem of calculus - der Hauptsatz der Differential- und Integralrechnung
The function f is a continuous function in the interval [a;b]. The function F is the antiderivative of f. Then the definite integral of f between a and b is calculated with the difference F(b) - F(a).
$$\int\limits_{a}^{b} f(x)\ dx = F(b) - F(a)$$
applications
example 1 - positive function
Let $$f(x) = \frac{1}{2}x$$ and you want to find the area in [1, 3]
After looking for a function F where F’ = f. F(b) is the area from 0 to b and and F(a) is the area from 0 to a. The difference F(b) - F(a) is then the area enclosed by f and the x-axis between a and b.
The antidervative of f is $F(x) = \frac{1}{4}x^2$.
$${\begin{align} \int\limits_{1}^{3} f(x)\ dx & = \left[\frac{1}{4}x^2 \right]_{1}^{3} \newline & = \frac{1}{4}\cdot 3^2 - \frac{1}{4}\cdot 1^2 \newline & = \frac{13}{2} \end{align}}$$
example 2 - negative function
Given $f(x) = (x-2)^2 - 1$. Find the area between the function and the x-axis.
$${\begin{align} \int\limits_{1}^{3} ((x-2)^2 - 1)\ dx & = \int\limits_{1}^{3} (x^2-4x + 3)\ dx \newline & = \left[\frac{1}{3}x^3 - \frac{4}{2}x^2 + 3x \right]_{\color{lime}{1}}^{\color{yellow}{3}} \newline & = \frac{1}{3}\cdot {\color{yellow}{3}}^3 - \frac{4}{2}\cdot {\color{yellow}{3}}^2 + 3 \cdot {\color{yellow}{3}} - (\frac{1}{3} \cdot {\color{lime}{1}}^3 - \frac{4}{2} \cdot {\color{lime}{1}}^2 + 3 \cdot {\color{lime}{1}}) \newline & = 9 - 18 + 9 -\frac{1}{3} + 2 - 3 \newline & = - \frac{4}{3} \end{align}}$$
Note: The roots are x=1 and x=3.
The result is negative, so to evaluate the area take the absolute value.
It follows
$$ \text{Area} = \left|{-\frac{4}{3}}\right| = \frac{4}{3} $$
See the function plotter below.
function plotter
Simple function plotter adapted from jsxgraph.org. Simply enter the function into the input box and click “plot”.
f(x) =
area as a sum of input and output - die FLächenbilanz
- Economy - balance sheet of income and expense
- Energy - energy budget, a balance sheet between energy income and expenditure.
The definite integral in an interval [a,b] is the balance in that interval.
how do we calculate the area when the function intersects the x-axis?
change of sign in the interval [a,b]
This has an effect on how we calculate the absolute area between the function f and the x-axis.
If there is a change of sign, you have to separate the integral into regions that are above the x-axis and regions that are below the x-axis. This is why it is important to be able to find the roots of a function.
Then take the absolute value of these regions.
$${\begin{align} \text{Area} & = \left| {\int\limits_{a}^{c} f(x)\ dx} \right| + \left| {\int\limits_{c}^{b} f(x)\ dx} \right| \end{align}}$$