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

##### 1 Answer

#### Answer:

The net area is

#### Explanation:

We seek the are under

**Method 1:**

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

# f(0) = -1 #

# f(2) = 9 #

and width

# 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 #