#### 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