# What is the area under f(x)=5x-1 in x in[0,2] ?

Mar 9, 2018

The net area is $8$, The actual area is $8.2 \setminus {\text{unit}}^{2}$

#### Explanation:

We seek the are under $f \left(x\right) = 5 x - 1$ where $x \in \left[0 , 2\right]$

Method 1:

The bounded net area is that of a trapezium with heights:

$f \left(0\right) = - 1$
$f \left(2\right) = 9$

and width $2$, So we can use the trapezium formula:

$A = \frac{1}{2} \left(a + b\right) h$
$\setminus \setminus \setminus = \frac{1}{2} \left(- 1 + 9\right) \left(2\right)$
$\setminus \setminus \setminus = 8$

Method 2:

We can use calculus, and evaluate the definite integral:

$A = {\int}_{a}^{b} \setminus f \left(x\right) \setminus \mathrm{dx}$
$\setminus \setminus \setminus = {\int}_{0}^{2} \setminus 5 x - 1 \setminus \mathrm{dx}$
$\setminus \setminus \setminus = {\left[\frac{5}{2} {x}^{2} - x\right]}_{0}^{2}$
$\setminus \setminus \setminus = \left(\frac{20}{2} - 2\right) - \left(0 - 0\right)$
$\setminus \setminus \setminus = 8$, as before

Note:

Both of the above methods calculate the "net" area, whereas the actual area is somewhat different:

graph{(y-5x+1)(y-10000x)(y-10000x+20000)=0 [-1, 3, -5, 12]}

The actual area is:

$A = \frac{1}{2} \left(\frac{1}{5}\right) \left(1\right) + \frac{1}{2} \left(\frac{9}{5}\right) \left(9\right)$
$\setminus \setminus \setminus = \frac{1}{10} + \frac{81}{10}$
$\setminus \setminus \setminus = \frac{82}{10}$
$\setminus \setminus \setminus = 8.2$