# AP Calculus BC : Length of Curve, Distance traveled, Accumulated Change, Motion of Curve

## Example Questions

### Example Question #1 : Integral Applications

Determine the length of the following function between

Explanation:

In order to begin the problem, we must first remember the formula for finding the arc length of a function along any given interval:

where ds is given by the equation below:

We can see from our equation for ds that we must find the derivative of our function, which in our case is dv/dt instead of dy/dx, so we begin by differentiating our function v(t) with respect to t:

Now we can plug this into the given equation to find ds:

Our last step is to plug our value for ds into the equation for arc length, which we can see only involves integrating ds. The interval along which the problem asks for the length of the function gives us our limits of integration, so we simply integrate ds from t=1 to t=4:

### Example Question #1 : Applications In Physics

In physics, the work done on an object is equal to the integral of the force on that object dotted with its displacent.

This looks like  ( is work,  is force, and  is the infinitesimally small displacement vector). For a force whose direction is the line of motion, the equation becomes .

If the force on an object as a function of displacement is , what is the work as a function of displacement ? Assume  and the force is in the direction of the object's motion.

Not enough information

Explanation:

, so .

Both the terms of the force are power terms in the form , which have the integral , so the integral of the force is .

We know

.

This means

.

### Example Question #11 : Application Of Integrals

Give the arclength of the graph of the function  on the interval .

Explanation:

The length of the curve of  on the interval  can be determined by evaluating the integral

.

so

.

The above integral becomes

Substitute . Then , and the integral becomes

### Example Question #1 : Length Of Curve, Distance Traveled, Accumulated Change, Motion Of Curve

Give the arclength of the graph of the function  on the interval .

Explanation:

The length of the curve of  on the interval  can be determined by evaluating the integral

.

, so

The integral becomes

Use substitution - set . Then , and . The bounds of integration become  and , and the integral becomes

### Example Question #1 : Average Values And Lengths Of Functions

What is the length of the curve  over the interval ?

Explanation:

The general formula for finding the length of a curve  over an interval  is

In this example, the arc length can be found by computing the integral

.

The derivative of  can be found using the power rule, , which leads to

At this point, a substitution is useful.

Let

.

We can also express the limits of integration in terms of  to simplify computation. When , and when .

Making these substitutions leads to

.

Now use the power rule, which in general is , to evaluate the integral.

### Example Question #1 : Length Of Curve, Distance Traveled, Accumulated Change, Motion Of Curve

Find the total distance traveled by a particle along the curve  from  to .