We have to distinguish distance from displacement. If we wanted to find the displacement of the object, we would take the definite integral of velocity and solve. However, when we solve for distance, we have to solve for the integral of 

We take the absolute value because distances can never be negative. 

Taking the integral of an absolute value function is difficult, instead we will do separate integrals for when  is positive and negative between  and .

  from  to . 

 from  to 

We can express the distance travelled  as

We split it up into two integrals because the sign of the integral would change, which is compensated for by the absolute value. 

Firstly, recall that

Also recall that

Knowing these two identites, we can solve for 

Solving for ,

There are two things we will need to solve for.  First is the vertex of the parabola, and second is where the ball will hit the ground.

Use the FOIL method to simplify the binomials of the height function.

Recall the vertex formula for the parabola to find the max.  Substitute the values of  and  since the height function is in standard form.

This is the position of the ball that is thrown from the skyscraper.

We will now need to know the roots of the function.  This is where the ball hits the ground.  Set the height function equal to zero and solve for the roots.

Since there is no such thing as negative distance, we will only consider the positive distance .  This is where the ball has hit the ground.

Subtract  from where the ball has been thrown from  where the ball has hit the ground.  This is the horizontal distance the ball has been thrown.

To solve, simply realize you were given d(t) which is the distance function. Thus, just plug in .

We know that distance is the difference of  positions, so we need a position function. We can find this by integrating the velocity function.

Recall to integrate this particular function we use, 

.

The car started at position , so , and we find

and . Our position function is now determined.

The distance the car traveled between  and  is the difference between the positions of the car at those  times. Hence, we have .

To solve, simply integrate  to find , the position function, and then plug in .

Velocity is the time derivative of position, and by that token position can be found by integrating a known velocity function with respect to time:

Now, if this integral were to be taken over an interval of time , this will give a finite value, a change in position, i.e. a distance travelled:

For the velocity function

The distance travelled can be found via knowledge of the following derivative properties:

Derivative of an exponential: 

The distance travelled over  is:

Pretty speedy.

Velocity is the time derivative of position, and by that token position can be found by integrating a known velocity function with respect to time:

Now, if this integral were to be taken over an interval of time , this will give a finite value, a change in position, i.e. a distance travelled:

For the velocity function

The distance travelled can be found via knowledge of the following derivative properties:

Trigonometric derivative: 

The distance travelled over  is:

Velocity is the time derivative of position, and by that token position can be found by integrating a known velocity function with respect to time:

Now, if this integral were to be taken over an interval of time , this will give a finite value, a change in position, i.e. a distance travelled:

For the velocity function

The distance travelled can be found via knowledge of the following derivative properties:

Trigonometric derivative: 

The distance travelled over  is:

The acceleration of the penny is equal to 10 m/s², meaning that every second that passes by, the velocity will increase by 10 m/s. In order to find a formula for the velocity of the penny, you must find the anti-derivative of the acceleration. Taking the anti-derivative of the veloctity will then give you a formula for the position of the penny. 

.

Since the initial velocity was zero, you can use this initial condition to solve for the constant C. Setting 10t + C equal to zero, C is solved to be equal to 0.

Therefore, .

This makes sense, since every second that passes by, the velocity will increase by 10.

.

The final integration (anti-derivative) yields this formula. Since we are only trying to find the distance traveled, and not its exact postion, the constant C is equal to zero.

Therefore, .

In order to find the total distance traveled, plug in the amount of time that passed into the postion formula you just found!

 meters traveled.

To find distance traveled, we will integrate the velocity function:

To integrate use the following rule,

Thus we get the following.

Then, plugging in t=20