Parametric Equations and Polar Coordinates

# 5 Area and Arc Length in Polar Coordinates

### Learning Objectives

• Apply the formula for area of a region in polar coordinates.
• Determine the arc length of a polar curve.

In the rectangular coordinate system, the definite integral provides a way to calculate the area under a curve. In particular, if we have a function defined from to where on this interval, the area between the curve and the x-axis is given by This fact, along with the formula for evaluating this integral, is summarized in the Fundamental Theorem of Calculus. Similarly, the arc length of this curve is given by In this section, we study analogous formulas for area and arc length in the polar coordinate system.

### Areas of Regions Bounded by Polar Curves

We have studied the formulas for area under a curve defined in rectangular coordinates and parametrically defined curves. Now we turn our attention to deriving a formula for the area of a region bounded by a polar curve. Recall that the proof of the Fundamental Theorem of Calculus used the concept of a Riemann sum to approximate the area under a curve by using rectangles. For polar curves we use the Riemann sum again, but the rectangles are replaced by sectors of a circle.

Consider a curve defined by the function where Our first step is to partition the interval into n equal-width subintervals. The width of each subinterval is given by the formula and the ith partition point is given by the formula Each partition point defines a line with slope passing through the pole as shown in the following graph.

A partition of a typical curve in polar coordinates.

The line segments are connected by arcs of constant radius. This defines sectors whose areas can be calculated by using a geometric formula. The area of each sector is then used to approximate the area between successive line segments. We then sum the areas of the sectors to approximate the total area. This approach gives a Riemann sum approximation for the total area. The formula for the area of a sector of a circle is illustrated in the following figure.

The area of a sector of a circle is given by

Recall that the area of a circle is When measuring angles in radians, 360 degrees is equal to radians. Therefore a fraction of a circle can be measured by the central angle The fraction of the circle is given by so the area of the sector is this fraction multiplied by the total area:

Since the radius of a typical sector in (Figure) is given by the area of the ith sector is given by

Therefore a Riemann sum that approximates the area is given by

We take the limit as to get the exact area:

This gives the following theorem.

Area of a Region Bounded by a Polar Curve

Suppose is continuous and nonnegative on the interval with The area of the region bounded by the graph of between the radial lines and is

Finding an Area of a Polar Region

Find the area of one petal of the rose defined by the equation

The graph of follows.

The graph of

When we have The next value for which is This can be seen by solving the equation for Therefore the values to trace out the first petal of the rose. To find the area inside this petal, use (Figure) with and

To evaluate this integral, use the formula with

Find the area inside the cardioid defined by the equation

Hint

Use (Figure). Be sure to determine the correct limits of integration before evaluating.

(Figure) involved finding the area inside one curve. We can also use (Figure) to find the area between two polar curves. However, we often need to find the points of intersection of the curves and determine which function defines the outer curve or the inner curve between these two points.

Finding the Area between Two Polar Curves

Find the area outside the cardioid and inside the circle

First draw a graph containing both curves as shown.

The region between the curves and

To determine the limits of integration, first find the points of intersection by setting the two functions equal to each other and solving for

This gives the solutions and which are the limits of integration. The circle is the red graph, which is the outer function, and the cardioid is the blue graph, which is the inner function. To calculate the area between the curves, start with the area inside the circle between and then subtract the area inside the cardioid between and

Find the area inside the circle and outside the circle

Hint

Use (Figure) and take advantage of symmetry.

In (Figure) we found the area inside the circle and outside the cardioid by first finding their intersection points. Notice that solving the equation directly for yielded two solutions: and However, in the graph there are three intersection points. The third intersection point is the origin. The reason why this point did not show up as a solution is because the origin is on both graphs but for different values of For example, for the cardioid we get

so the values for that solve this equation are where n is any integer. For the circle we get

The solutions to this equation are of the form for any integer value of n. These two solution sets have no points in common. Regardless of this fact, the curves intersect at the origin. This case must always be taken into consideration.

### Arc Length in Polar Curves

Here we derive a formula for the arc length of a curve defined in polar coordinates.

In rectangular coordinates, the arc length of a parameterized curve for is given by

In polar coordinates we define the curve by the equation where In order to adapt the arc length formula for a polar curve, we use the equations

and we replace the parameter t by Then

We replace by and the lower and upper limits of integration are and respectively. Then the arc length formula becomes

This gives us the following theorem.

Arc Length of a Curve Defined by a Polar Function

Let be a function whose derivative is continuous on an interval The length of the graph of from to is

Finding the Arc Length of a Polar Curve

Find the arc length of the cardioid

When Furthermore, as goes from to the cardioid is traced out exactly once. Therefore these are the limits of integration. Using and (Figure) becomes

Next, using the identity add 1 to both sides and multiply by 2. This gives Substituting gives so the integral becomes

The absolute value is necessary because the cosine is negative for some values in its domain. To resolve this issue, change the limits from to and double the answer. This strategy works because cosine is positive between and Thus,

Find the total arc length of

Hint

Use (Figure). To determine the correct limits, make a table of values.

### Key Concepts

• The area of a region in polar coordinates defined by the equation with is given by the integral
• To find the area between two curves in the polar coordinate system, first find the points of intersection, then subtract the corresponding areas.
• The arc length of a polar curve defined by the equation with is given by the integral

### Key Equations

• Area of a region bounded by a polar curve
• Arc length of a polar curve

For the following exercises, determine a definite integral that represents the area.

Region enclosed by

Region enclosed by

Region in the first quadrant within the cardioid

Region enclosed by one petal of

Region enclosed by one petal of

Region below the polar axis and enclosed by

Region in the first quadrant enclosed by

Region enclosed by the inner loop of

Region enclosed by the inner loop of

Region enclosed by and outside the inner loop

Region common to

Region common to

Region common to

For the following exercises, find the area of the described region.

Enclosed by

Above the polar axis enclosed by

Below the polar axis and enclosed by

Enclosed by one petal of

Enclosed by one petal of

Enclosed by

Enclosed by the inner loop of

Enclosed by and outside the inner loop

Common interior of

Common interior of

Common interior of

Inside and outside

Common interior of

For the following exercises, find a definite integral that represents the arc length.

on the interval

For the following exercises, find the length of the curve over the given interval.

32

For the following exercises, use the integration capabilities of a calculator to approximate the length of the curve.

[T]

6.238

[T]

[T]

2

[T]

[T]

4.39

For the following exercises, use the familiar formula from geometry to find the area of the region described and then confirm by using the definite integral.

For the following exercises, use the familiar formula from geometry to find the length of the curve and then confirm using the definite integral.

Verify that if then

For the following exercises, find the slope of a tangent line to a polar curve Let and so the polar equation is now written in parametric form.

Use the definition of the derivative and the product rule to derive the derivative of a polar equation.

The slope is

The slope is 0.

tips of the leaves

At the slope is undefined. At the slope is 0.

tips of the leaves

The slope is undefined at

Find the points on the interval at which the cardioid has a vertical or horizontal tangent line.

For the cardioid find the slope of the tangent line when

Slope = −1.

For the following exercises, find the slope of the tangent line to the given polar curve at the point given by the value of

Slope is

[T] Use technology: at

For the following exercises, find the points at which the following polar curves have a horizontal or vertical tangent line.

Horizontal tangent at

The cardioid

Horizontal tangents at Vertical tangents at and also at the pole

Show that the curve (called a cissoid of Diocles) has the line as a vertical asymptote.