To determine

To represent:

What does ∫35fxdx  represent.

Answer

Represents the difference in altitude of the trail between x= 3 miles and x=5 miles.

Explanation

1) Concept:

The Net Change Theorem: The integral of a rate of change is the net change.

∫abF'xdx=Fb-Fa

2) Given:

∫35fxdx 

3) Calculation:

Consider the integral,

∫35fxdx , where fx represents the slope of a trail at a distance of x miles from the start of the trail.

Let  fx=F'x; F(x) be the distance function of the trail.

 Substitute the value fx=F'x in ∫35fxdx .∫35fxdx=∫35F'xdx

By using the Net Change Theorem,

∫35F'xdx=F5-F3

∫35fxdx =F5-F3

The slope of trail is rate of change of altitude. Thus it follows that F(x) is the altitude of the trail at x. So the integral ∫35fxdx  represents the difference in altitude of the trail between x= 3 miles and x=5 miles.

Conclusion:

Therefore, the integral ∫35fxdx  represents the difference in altitude of the trail between x= 3 miles and x=5 miles.s