Parametric Equations and Polar Coordinates

# 3 Calculus of Parametric Curves

### Learning Objectives

• Determine derivatives and equations of tangents for parametric curves.
• Find the area under a parametric curve.
• Use the equation for arc length of a parametric curve.
• Apply the formula for surface area to a volume generated by a parametric curve.

Now that we have introduced the concept of a parameterized curve, our next step is to learn how to work with this concept in the context of calculus. For example, if we know a parameterization of a given curve, is it possible to calculate the slope of a tangent line to the curve? How about the arc length of the curve? Or the area under the curve?

Another scenario: Suppose we would like to represent the location of a baseball after the ball leaves a pitcher’s hand. If the position of the baseball is represented by the plane curve then we should be able to use calculus to find the speed of the ball at any given time. Furthermore, we should be able to calculate just how far that ball has traveled as a function of time.

### Derivatives of Parametric Equations

We start by asking how to calculate the slope of a line tangent to a parametric curve at a point. Consider the plane curve defined by the parametric equations

The graph of this curve appears in (Figure). It is a line segment starting at and ending at

Graph of the line segment described by the given parametric equations.

We can eliminate the parameter by first solving the equation for t:

Substituting this into we obtain

The slope of this line is given by Next we calculate and This gives and Notice that This is no coincidence, as outlined in the following theorem.

Derivative of Parametric Equations

Consider the plane curve defined by the parametric equations and Suppose that and exist, and assume that Then the derivative is given by

#### Proof

This theorem can be proven using the Chain Rule. In particular, assume that the parameter t can be eliminated, yielding a differentiable function Then Differentiating both sides of this equation using the Chain Rule yields

so

But which proves the theorem.

(Figure) can be used to calculate derivatives of plane curves, as well as critical points. Recall that a critical point of a differentiable function is any point such that either or does not exist. (Figure) gives a formula for the slope of a tangent line to a curve defined parametrically regardless of whether the curve can be described by a function or not.

Finding the Derivative of a Parametric Curve

Calculate the derivative for each of the following parametrically defined plane curves, and locate any critical points on their respective graphs.

1. To apply (Figure), first calculate and

Next substitute these into the equation:

This derivative is undefined when Calculating and gives and which corresponds to the point on the graph. The graph of this curve is a parabola opening to the right, and the point is its vertex as shown.

Graph of the parabola described by parametric equations in part a.

2. To apply (Figure), first calculate and

Next substitute these into the equation:

This derivative is zero when When we have

which corresponds to the point on the graph. When we have

which corresponds to the point on the graph. The point is a relative minimum and the point is a relative maximum, as seen in the following graph.

Graph of the curve described by parametric equations in part b.

3. To apply (Figure), first calculate and

Next substitute these into the equation:

This derivative is zero when and is undefined when This gives as critical points for t. Substituting each of these into and we obtain

0 5 0
0 5
−5 0
0 −5
5 0

These points correspond to the sides, top, and bottom of the circle that is represented by the parametric equations ((Figure)). On the left and right edges of the circle, the derivative is undefined, and on the top and bottom, the derivative equals zero.

Graph of the curve described by parametric equations in part c.

Calculate the derivative for the plane curve defined by the equations

and locate any critical points on its graph.

and so
This expression is undefined when and equal to zero when

Hint

Calculate and and use (Figure).

Finding a Tangent Line

Find the equation of the tangent line to the curve defined by the equations

First find the slope of the tangent line using (Figure), which means calculating and

Next substitute these into the equation:

When so this is the slope of the tangent line. Calculating and gives

which corresponds to the point on the graph ((Figure)). Now use the point-slope form of the equation of a line to find the equation of the tangent line:

Tangent line to the parabola described by the given parametric equations when

Find the equation of the tangent line to the curve defined by the equations

The equation of the tangent line is

Hint

Calculate and and use (Figure).

### Second-Order Derivatives

Our next goal is to see how to take the second derivative of a function defined parametrically. The second derivative of a function is defined to be the derivative of the first derivative; that is,

Since we can replace the on both sides of this equation with This gives us

If we know as a function of t, then this formula is straightforward to apply.

Finding a Second Derivative

Calculate the second derivative for the plane curve defined by the parametric equations

From (Figure) we know that Using (Figure), we obtain

Calculate the second derivative for the plane curve defined by the equations

and locate any critical points on its graph.

Critical points

Hint

### Integrals Involving Parametric Equations

Now that we have seen how to calculate the derivative of a plane curve, the next question is this: How do we find the area under a curve defined parametrically? Recall the cycloid defined by the equations Suppose we want to find the area of the shaded region in the following graph.

Graph of a cycloid with the arch over highlighted.

To derive a formula for the area under the curve defined by the functions

we assume that is differentiable and start with an equal partition of the interval Suppose and consider the following graph.

Approximating the area under a parametrically defined curve.

We use rectangles to approximate the area under the curve. The height of a typical rectangle in this parametrization is for some value in the ith subinterval, and the width can be calculated as Thus the area of the ith rectangle is given by

Then a Riemann sum for the area is

Multiplying and dividing each area by gives

Taking the limit as approaches infinity gives

This leads to the following theorem.

Area under a Parametric Curve

Consider the non-self-intersecting plane curve defined by the parametric equations

and assume that is differentiable. The area under this curve is given by

Finding the Area under a Parametric Curve

Find the area under the curve of the cycloid defined by the equations

Using (Figure), we have

Find the area under the curve of the hypocycloid defined by the equations

(Note that the integral formula actually yields a negative answer. This is due to the fact that is a decreasing function over the interval that is, the curve is traced from right to left.)

Hint

Use (Figure), along with the identities and

### Arc Length of a Parametric Curve

In addition to finding the area under a parametric curve, we sometimes need to find the arc length of a parametric curve. In the case of a line segment, arc length is the same as the distance between the endpoints. If a particle travels from point A to point B along a curve, then the distance that particle travels is the arc length. To develop a formula for arc length, we start with an approximation by line segments as shown in the following graph.

Approximation of a curve by line segments.

Given a plane curve defined by the functions we start by partitioning the interval into n equal subintervals: The width of each subinterval is given by We can calculate the length of each line segment:

Then add these up. We let s denote the exact arc length and denote the approximation by n line segments:

If we assume that and are differentiable functions of t, then the Mean Value Theorem (Introduction to the Applications of Derivatives) applies, so in each subinterval there exist and such that

Therefore (Figure) becomes

This is a Riemann sum that approximates the arc length over a partition of the interval If we further assume that the derivatives are continuous and let the number of points in the partition increase without bound, the approximation approaches the exact arc length. This gives

When taking the limit, the values of and are both contained within the same ever-shrinking interval of width so they must converge to the same value.

We can summarize this method in the following theorem.

Arc Length of a Parametric Curve

Consider the plane curve defined by the parametric equations

and assume that and are differentiable functions of t. Then the arc length of this curve is given by

At this point a side derivation leads to a previous formula for arc length. In particular, suppose the parameter can be eliminated, leading to a function Then and the Chain Rule gives Substituting this into (Figure) gives

Here we have assumed that which is a reasonable assumption. The Chain Rule gives and letting and we obtain the formula

which is the formula for arc length obtained in the Introduction to the Applications of Integration.

Finding the Arc Length of a Parametric Curve

Find the arc length of the semicircle defined by the equations

The values to trace out the red curve in (Figure). To determine its length, use (Figure):

Note that the formula for the arc length of a semicircle is and the radius of this circle is 3. This is a great example of using calculus to derive a known formula of a geometric quantity.

The arc length of the semicircle is equal to its radius times

Find the arc length of the curve defined by the equations

Hint

Use (Figure).

We now return to the problem posed at the beginning of the section about a baseball leaving a pitcher’s hand. Ignoring the effect of air resistance (unless it is a curve ball!), the ball travels a parabolic path. Assuming the pitcher’s hand is at the origin and the ball travels left to right in the direction of the positive x-axis, the parametric equations for this curve can be written as

where t represents time. We first calculate the distance the ball travels as a function of time. This distance is represented by the arc length. We can modify the arc length formula slightly. First rewrite the functions and using v as an independent variable, so as to eliminate any confusion with the parameter t:

Then we write the arc length formula as follows:

The variable v acts as a dummy variable that disappears after integration, leaving the arc length as a function of time t. To integrate this expression we can use a formula from Appendix A,

We set and This gives so Therefore

and

This function represents the distance traveled by the ball as a function of time. To calculate the speed, take the derivative of this function with respect to t. While this may seem like a daunting task, it is possible to obtain the answer directly from the Fundamental Theorem of Calculus:

Therefore

One third of a second after the ball leaves the pitcher’s hand, the distance it travels is equal to

This value is just over three quarters of the way to home plate. The speed of the ball is

This speed translates to approximately 95 mph—a major-league fastball.

### Surface Area Generated by a Parametric Curve

Recall the problem of finding the surface area of a volume of revolution. In Curve Length and Surface Area, we derived a formula for finding the surface area of a volume generated by a function from to revolved around the x-axis:

We now consider a volume of revolution generated by revolving a parametrically defined curve around the x-axis as shown in the following figure.

A surface of revolution generated by a parametrically defined curve.

The analogous formula for a parametrically defined curve is

provided that is not negative on

Finding Surface Area

Find the surface area of a sphere of radius r centered at the origin.

This generates an upper semicircle of radius r centered at the origin as shown in the following graph.

A semicircle generated by parametric equations.

When this curve is revolved around the x-axis, it generates a sphere of radius r. To calculate the surface area of the sphere, we use (Figure):

This is, in fact, the formula for the surface area of a sphere.

Find the surface area generated when the plane curve defined by the equations

is revolved around the x-axis.

Hint

Use (Figure). When evaluating the integral, use a u-substitution.

### Key Concepts

• The derivative of the parametrically defined curve and can be calculated using the formula Using the derivative, we can find the equation of a tangent line to a parametric curve.
• The area between a parametric curve and the x-axis can be determined by using the formula
• The arc length of a parametric curve can be calculated by using the formula
• The surface area of a volume of revolution revolved around the x-axis is given by If the curve is revolved around the y-axis, then the formula is

### Key Equations

• Derivative of parametric equations
• Second-order derivative of parametric equations
• Area under a parametric curve
• Arc length of a parametric curve
• Surface area generated by a parametric curve

For the following exercises, each set of parametric equations represents a line. Without eliminating the parameter, find the slope of each line.

0

For the following exercises, determine the slope of the tangent line, then find the equation of the tangent line at the given value of the parameter.

Slope is undefined;

For the following exercises, find all points on the curve that have the given slope.

slope = 0.5

No points possible; undefined expression.

For the following exercises, write the equation of the tangent line in Cartesian coordinates for the given parameter t.

For where Find all values of t at which a horizontal tangent line exists.

For where Find all values of t at which a vertical tangent line exists.

Find all points on the curve that have the slope of

Find for

Find the equation of the tangent line to at

For the curve find the slope and concavity of the curve at

and so the curve is neither concave up nor concave down at Therefore the graph is linear and has a constant slope but no concavity.

For the parametric curve whose equation is find the slope and concavity of the curve at

Find the slope and concavity for the curve whose equation is at

the curve is concave down at

Find all points on the curve at which there are vertical and horizontal tangents.

Find all points on the curve at which horizontal and vertical tangents exist.

No horizontal tangents. Vertical tangents at

For the following exercises, find

For the following exercises, find points on the curve at which tangent line is horizontal or vertical.

Horizontal vertical

For the following exercises, find at the value of the parameter.

1

0

For the following exercises, find at the given point without eliminating the parameter.

4

Find t intervals on which the curve is concave up as well as concave down.

Determine the concavity of the curve

Concave up on

Sketch and find the area under one arch of the cycloid

Find the area bounded by the curve and the lines and

1

Find the area enclosed by the ellipse

Find the area of the region bounded by for

For the following exercises, find the area of the regions bounded by the parametric curves and the indicated values of the parameter.

[T]

[T] (the “hourglass”)

[T] (the “teardrop”)

For the following exercises, find the arc length of the curve on the indicated interval of the parameter.

(express answer as a decimal rounded to three places)

on the interval (the hypocycloid)

Find the length of one arch of the cycloid

Find the distance traveled by a particle with position as t varies in the given time interval:

Find the length of one arch of the cycloid

Show that the total length of the ellipse is where and

Find the length of the curve

For the following exercises, find the area of the surface obtained by rotating the given curve about the x-axis.

[T] Use a CAS to find the area of the surface generated by rotating about the x-axis. (Answer to three decimal places.)

59.101

Find the surface area obtained by rotating about the y-axis.

Find the area of the surface generated by revolving about the x-axis.

Find the surface area generated by revolving about the y-axis.