Consider the functions f$($x$)=$ x$^3-4$x and g$($x$)=$ x$.$ Determine where the two functions intersect by solving the equation:

f$($x$)=$ g$($x$)$ x$^3-4$x $=$ x $$ x$^3-5$x $=0$ x$($x$^2-5)=0$

So$,$ the points of intersection are x $=0$ and x $=$ sqrt$(5),-$sqrt$(5).$ Choose the interval from x $=0$ to x $=$ sqrt$(5).$

To find the area between curves, use the formula:

Area $=$ from a to b of $|$f$($x$)-$ g$($x$)|$ dx

So$,$ Area $=$ from $0$ to sqrt$(5)$ of $|($x$^3-4$x $-$ x$)|$ dx

$=$ from $0$ to sqrt$(5)$ of $|$x$^3-5$x$|$ dx

Evaluate this integral step by step and compute the total area.

Next, find the average value of the function h$($x$)=$ f$($x$)-$ g$($x$)$ over the same interval using:

Average $=(1/($b $-$ a$))$ from a to b of h$($x$)$ dx

Then, find the value of x at which this average is attained.

Now compute the centroid using standard formulas:

x$=(1/$A$)$ from a to b of x$($f$($x$)-$ g$($x$))$ dx

y$=(1/2$A$)$ from a to b of $($f$($x$)^2-$ g$($x$)^2)$ dx

Sketch both f$($x$)$ and g$($x$),$ shade the area, and label intersection points.

Explain how signed area differs from geometric area and give two real$-$world uses.