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

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Answer 1 · 2018-03-09T21:10:11

The net area is $8$ , The actual area is $8.2 \ "unit"^2$

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

Method 1:

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

$f(0) = -1$
$f(2) = 9$

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

$A=1/2(a+b)h$
$\ \ \ =1/2(-1+9)(2)$
$\ \ \ =8$

Method 2:

We can use calculus, and evaluate the definite integral:

$A =int_a^b \ f(x) \ dx$
$\ \ \ =int_0^2 \ 5x-1 \ dx$
$\ \ \ =[5/2x^2-x]_0^2$
$\ \ \ =(20/2-2)-(0-0)$
$\ \ \ =8$ , 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:

$A = 1/2(1/5)(1) + 1/2(9/5)(9)$
$\ \ \ = 1/10 + 81/10$
$\ \ \ = 82/10$
$\ \ \ = 8.2$

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