Question

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

Answer 1

`86/3`

Explanation:

The integral of the function `f(x)` on the interval `x in[a,b]`, that is, `int_a^bf(x)dx` is equal to the area between the curve and the `x`-axis. Thus, the answer to your question can be written as

`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, `8`, can be integrated using the following rule where `k` is a constant:

`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 `f(x)` evaluated from `2` to `4`, we must take `F(4)-F(2)` (this is the first fundamental theorem of calculus).

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`