# What is the net area between #f(x) = x^2-x+8 # and the x-axis over #x in [2, 4 ]#?

##### 1 Answer

#### Explanation:

The integral of the function

#int_2^4(x^2-x+8)dx#

We can integrate the first two terms using the following rule of integration:

#intx^n=x^(n+1)/(n+1)+C#

The last,

#intkdx=kx+C#

So, when we integrate this function (momentarily ignoring the integral's bounds), we get the antiderivative

#F(x)=x^3/3-x^2/2+8x+C#

So, to find the integral of

This can be notated as

#int_2^4(x^2-x+8)dx=[x^3/3-x^2/2+8x]_2^4#

#=(4^3/3-4^2/2+8(4))-(2^3/3-2^2/2+8(2))#

#=(64/3-8+32)-(8/3-2+16)#

#=(136/3)-(50/3)=color(blue)(86/3#