Suppose a we drive a car 70 miles down a straight road in two hours.

Then we return to our starting point in another two hours.

A graph of the trip in terms of velocity versus time is shown.

The displacement of the car is given by integrating the area under the curve over a particular time period.

During the first two hours, the displacement of the car is 35 miles.

During the time interval from two hours to four hours, the displacement of the car is -35 miles.

The displacement, or net change of the car's position, is zero miles.

The car travels 35 miles during the first two hours, and another 35 miles on the return during the next two hours, making a total distance traveled equal to 70 miles.

We see that the total distance traveled does not depend on the sign of the velocity function. It only depends on the speed, which is the absolute value of the velocity.

Therefore, the total distance traveled is given by the integral of the absolute value of the velocity function.

Consider this velocity function of an object moving along the x-axis.

The graph of the velocity function is shown. The position of the object is shown on the x-axis shown below it.

Move the slider to change the time t. Notice the position of the object on the x-axis.

The object stops when the velocity is zero, and then moves in the reverse direction as the velocity becomes negative.

A ball is shot in a straight line vertically upward.

Its acceleration is constant as shown by the graph.

If we know the initial velocity of the ball, then we can determine the constant of integration.

The graph of the velocity function is a straight line.

We can use the velocity function to determine the time at which the ball changes direction. It is the time at which the velocity is zero.

We can determine the vertical position of the ball by calculating its displacement over the same time interval.

What is the displacement of the car after 4 hours? Click the "Submit" button after selecting your answer.

What is the total distance traveled by the car after 4 hours? Click the "Submit" button after selecting your answer.

What is the total distance traveled by the object during the time interval zero to 5 seconds? Click the "Submit" button after entering your answer.

At what value of t does the object change direction? Click the "Submit" button after entering your answer.

Which of the following is an equation for the velocity of the ball? Click the "Submit" button after selecting your answer.

Now, what is the total distance traveled by the ball after 16 seconds? Click the "Submit" button after entering your answer.

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