Integral as Net Change

A honey bee makes several trips from the hive to a flower garden. The velocity graph is shown below. What is the total distance traveled by the bee? 700 feet 200ft 200ft 200ft 100ft

What is the displacement of the bee? 100 feet towards the hive 200ft 200ft -200ft -100ft

To find the displacement (position shift) from the velocity function, we just integrate the function. The negative areas below the x-axis subtract from the total displacement. To find distance traveled we have to use absolute value. Find the roots of the velocity equation and integrate in pieces, just like when we found the area between a curve and the x-axis. (Take the absolute value of each integral.) Or you can use your calculator to integrate the absolute value of the velocity function.

Displacement: Distance Traveled: velocity graph position graph Every AP exam I have seen has had at least one problem requiring students to interpret velocity and position graphs.

In the linear motion equation: V(t) is a function of time. For a very small change in time, V(t) can be considered a constant. We add up all the small changes in S to get the total distance.

We add up all the small changes in S to get the total distance. As the number of subintervals becomes infinitely large (and the width becomes infinitely small), we have integration.

This same technique is used in many different real-life problems.

Work: Calculating the work is easy when the force and distance are constant. When the amount of force varies, we get to use calculus!

x = distance that the spring is extended beyond its natural length Hooke’s law for springs: k = spring constant x = distance that the spring is extended beyond its natural length

Hooke’s law for springs: F=10 N Example 7: x=2 M It takes 10 Newtons to stretch a spring 2 meters beyond its natural length. How much work is done stretching the spring to 4 meters beyond its natural length?

F(x) How much work is done stretching the spring to 4 meters beyond its natural length? x=4 M For a very small change in x, the force is constant. newton-meters joules p