To find total distance, integrate the absolute value of the velocity function over the specified interval. The easiest way to do this is to break the integral into two separate pieces, one for the region where sin(t) is positive and one for the region where sin(t) is negative.

Recall that sin(t) is positive on the interval from 0 to . The distance travelled over this interval is:

Next, compute the integral of sin(t) over the interval . Note that sin(t) is negative on this interval, so the value of the integral will be negative. After evaluating the integral, take the absolute value of the answer to find the distance travelled. 

 , so the distance traveled is 1.

Putting the distances found from the two integrals together, we find that the total distance traveled is 2+1 = 3.

To compute the change of distance using the velocity equation, you need to take the definite integral of the velocity function.  This would be:

This will require you to evaluate the following function:

, Thus you have:

Remember that the velcoity function can be found by taking the integral of the acceleration function.  Likewise, the position can be found by taking the integral of the velocity function.  We know:

Therefore, we know:

However,  must be , given the initial velocity of the particle.  This means that the distance can be computed:

This means you need to evaluate  at both  and :

The position function for a physical body can be computed using the integral of the velocity function:

Thus, we know that 

.  

Since, at , the position is , we know that .  

Therefore, we can solve for :

Based on our data, we know that the equation for the acceleration of the ship is:

Now, we know that we can find the velocity of the ship by using an integral so that:

This means:

However, since we know that at time , the velocity was ,  must equal .  Therefore, we know:

Now, the distance traveled by the ship during this period can be found by taking the definite integral from  to  of :

Therefore, we need to evaluate the following function for  and :

In order to solve this problem, you must know that the derivative of a position function is the veloctiy function and the derivative fo the velocity function is the acceleration function.  Thefore, by taking the double integral of the acceleration function we will be able to find the position function.  

In order to take the integral of the acceleration function, , we must first know the general rule of integration .  

Applying the general rule of integration, we find the velocity function to be .  Plugging in the known value of the intial velocity of the cannonball, we find that .  We also find that the moment in time that the velocity of the cannonball is equal to 0 is at 6 seconds.  This is the moment in time the cannonball stops moving, we will need to plug this into the position function to find how far the cannonball traveled.

Taking the integral of the veloctiy function with the general rule of integration, we find that position function becomes .  Since the intial position of the cannonball is 0, the position function becomes .  

Now by plugging in 6 seconds, we can find how far the ball traveled, which in this case is .

The distance a particle travels during an interval of time  can be found by integrating the velocity function over that interval:

If velocity is given as:

Then the distance travelled in three seconds from the beginning of movement is:

Distance travelled can be found by integrating the velocity function over the interval of elapsed time :

If velocity is:

Then the distance requested in this problem is:

This integral can be found using the method of integration by parts:

Now, use the method of integration by parts once more, on the rightmost integral:

Now, with  terms on the left and right side of the equation, we can move the right-side term to the left to get:

Now, we can find at last the distance travelled:

To find the position of the object after a certain time, we first must find the position function of the rocket, then solve the position function at that given time.

The position function of an object moving with uniform acceleration is , where  is the inital position of the object,  is the inital velocity of the object and  is the acceleration of the object.

For this problem:

, , , 

After  seconds, the rocket will be at a height of .

Distance a particle travels can be found by integrating the velocity function over a specified interval with respect to time. For distance traveled from time a to time b:

For the velocity function

Since we're looking at a distance traveled from an initial position, we'll assume a starting time of zero . Furthermore, since we want to know the time that the position reaches its initial position, we'll want a total distance of travel of zero :

For a postive value of time, the particle will return to its original postion at a time of