Learning Objectives
- 7.4.1 Apply the formula for area of a region in polar coordinates.
- 7.4.2 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 y=f(x) defined from x=a to x=b where f(x)>0 on this interval, the area between the curve and the x-axis is given by A=∫baf(x)dx. 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 L=∫ba√1+(f′(x))2dx. 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 r=f(θ), where α≤θ≤β. Our first step is to partition the interval [α,β] into n equal-width subintervals. The width of each subinterval is given by the formula Δθ=(β−α)/n, and the ith partition point θi is given by the formula θi=α+iΔθ. Each partition point θ=θi defines a line with slope tanθi passing through the pole as shown in the following graph.
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.
Recall that the area of a circle is A=πr2. When measuring angles in radians, 360 degrees is equal to 2π radians. Therefore a fraction of a circle can be measured by the central angle θ. The fraction of the circle is given by θ2π, so the area of the sector is this fraction multiplied by the total area:
Since the radius of a typical sector in Figure 7.39 is given by ri=f(θi), 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 n→∞ to get the exact area:
This gives the following theorem.
Theorem 7.6
Area of a Region Bounded by a Polar Curve
Suppose f is continuous and nonnegative on the interval α≤θ≤β with 0<β−α≤2π. The area of the region bounded by the graph of r=f(θ) between the radial lines θ=α and θ=β is
Example 7.16
Finding an Area of a Polar Region
Find the area of one petal of the rose defined by the equation r=3sin(2θ).
Solution
The graph of r=3sin(2θ) follows.
When θ=0 we have r=3sin(2(0))=0. The next value for which r=0 is θ=π/2. This can be seen by solving the equation 3sin(2θ)=0 for θ. Therefore the values θ=0 to θ=π/2 trace out the first petal of the rose. To find the area inside this petal, use Equation 7.9 with f(θ)=3sin(2θ), α=0, and β=π/2:
To evaluate this integral, use the formula sin2α=(1−cos(2α))/2 with α=2θ:
Checkpoint 7.15
Find the area inside the cardioid defined by the equation r=1−cosθ.
Example 7.16 involved finding the area inside one curve. We can also use Area of a Region Bounded by a Polar Curve 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.
Example 7.17
Finding the Area between Two Polar Curves
Find the area outside the cardioid r=2+2sinθ and inside the circle r=6sinθ.
Solution
First draw a graph containing both curves as shown.
To determine the limits of integration, first find the points of intersection by setting the two functions equal to each other and solving for θ:
This gives the solutions θ=π6 and θ=5π6, which are the limits of integration. The circle r=3sinθ is the red graph, which is the outer function, and the cardioid r=2+2sinθ is the blue graph, which is the inner function. To calculate the area between the curves, start with the area inside the circle between θ=π6 and θ=5π6, then subtract the area inside the cardioid between θ=π6 and θ=5π6:
Checkpoint 7.16
Find the area inside the circle r=4cosθ and outside the circle r=2.
In Example 7.17 we found the area inside the circle and outside the cardioid by first finding their intersection points. Notice that solving the equation directly for θ yielded two solutions: θ=π6 and θ=5π6. 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 θ. For example, for the cardioid we get
so the values for θ that solve this equation are θ=3π2+2nπ, where n is any integer. For the circle we get
The solutions to this equation are of the form θ=nπ 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 (x(t),y(t)) for a≤t≤b is given by
In polar coordinates we define the curve by the equation r=f(θ), where α≤θ≤β. In order to adapt the arc length formula for a polar curve, we use the equations
and we replace the parameter t by θ. Then
We replace dt by dθ, and the lower and upper limits of integration are α and β, respectively. Then the arc length formula becomes
This gives us the following theorem.
Theorem 7.7
Arc Length of a Curve Defined by a Polar Function
Let f be a function whose derivative is continuous on an interval α≤θ≤β. The length of the graph of r=f(θ) from θ=α to θ=β is
Example 7.18
Finding the Arc Length of a Polar Curve
Find the arc length of the cardioid r=2+2cosθ.
Solution
When θ=0,r=2+2cos0=4. Furthermore, as θ goes from 0 to 2π, the cardioid is traced out exactly once. Therefore these are the limits of integration. Using f(θ)=2+2cosθ, α=0, and β=2π, Equation 7.10 becomes
Next, using the identity cos(2α)=2cos2α−1, add 1 to both sides and multiply by 2. This gives 2+2cos(2α)=4cos2α. Substituting α=θ/2 gives 2+2cosθ=4cos2(θ/2), 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 0 to π and double the answer. This strategy works because cosine is positive between 0 and π2. Thus,
Checkpoint 7.17
Find the total arc length of r=3sinθ.
Section 7.4 Exercises
For the following exercises, determine a definite integral that represents the area.
Region enclosed by r=4
Region in the first quadrant within the cardioid r=1+sinθ
Region enclosed by one petal of r=cos(3θ)
Region in the first quadrant enclosed by r=2−cosθ
Region enclosed by the inner loop of r=3−4cosθ
Region common to r=3sinθandr=2−sinθ
Region common to r=3cosθandr=3sinθ
For the following exercises, find the area of the described region.
Above the polar axis enclosed by r=2+sinθ
Enclosed by one petal of r=4cos(3θ)
Enclosed by r=1+sinθ
Enclosed by r=2+4cosθ and outside the inner loop
Common interior of r=3−2sinθandr=−3+2sinθ
Inside r=1+cosθ and outside r=cosθ
For the following exercises, find a definite integral that represents the arc length.
r=4cosθon the interval0≤θ≤π2
r=2secθon the interval0≤θ≤π3
For the following exercises, find the length of the curve over the given interval.
r=6on the interval0≤θ≤π2
r=6cosθon the interval0≤θ≤π2
r=1−sinθon the interval0≤θ≤2π
For the following exercises, use the integration capabilities of a calculator to approximate the length of the curve.
[T] r=2θon the intervalπ≤θ≤2π
[T] r=2θ2on the interval0≤θ≤π
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.
r=3sinθon the interval0≤θ≤π
r=6sinθ+8cosθon the interval0≤θ≤π
For the following exercises, use the familiar formula from geometry to find the length of the curve and then confirm using the definite integral.
r=sinθ+cosθon the interval0≤θ≤π
Verify that if y=rsinθ=f(θ)sinθ then dydθ=f′(θ)sinθ+f(θ)cosθ.
For the following exercises, find the slope of a tangent line to a polar curve r=f(θ). Let x=rcosθ=f(θ)cosθ and y=rsinθ=f(θ)sinθ, so the polar equation r=f(θ) is now written in parametric form.
Use the definition of the derivative dydx=dy/dθdx/dθ and the product rule to derive the derivative of a polar equation.
r=1−sinθ; (12,π6)
r=8sinθ; (4,5π6)
r=6+3cosθ; (3,π)
r=2sin(3θ); tips of the leaves
Find the points on the interval −π≤θ≤π at which the cardioid r=1−cosθ has a vertical or horizontal tangent line.
For the following exercises, find the slope of the tangent line to the given polar curve at the point given by the value of θ.
r=3cosθ,θ=π3
r=lnθ, θ=e
For the following exercises, find the points at which the following polar curves have a horizontal or vertical tangent line.
r=4cosθ
r=2sin(2θ)
Show that the curve r=sinθtanθ (called a cissoid of Diocles) has the line x=1 as a vertical asymptote.