11.1.1How small would you like your slice of pi?

Area Bounded by a Polar Curve

You have calculated area bounded by many different curves, but how about polar curves? Today you will develop a method to calculate the area bounded by a polar curve.

11-1.

A sector is a portion of the interior of a circle bounded by two radii and the intercepted arc. Calculate the area of the sector shown at right.

Portion of a circle, central angle labeled, 1 sixth pi radians, & radius labeled, r = 4.2 cm.

11-2.

Explore how to calculate the area within a polar curve. What is your best estimate of the area of the region bounded by the limaçon $r = 7 + 6\cos(θ)$ , shown at right?

The area of the region bounded by the limaçon may be approximated by slicing the region into pie-shaped slices, estimating the area of each slice, and then adding the areas of the individual slices. Using the Lesson 11.1.1 Resource Page, divide the region into pie-shaped slices. Each slice should have a vertex at the pole (origin), and the angle measurements on the resource page are given in increments of $\frac { \pi } { 12 }$ radians.

Limacon without inner loops, with approximate turning points as follows: (negative 1, comma 0), (negative 2, comma 3), (5, comma 9), (13, comma 0), (5, comma negative 9), (negative 2, comma negative 3), & back to (negative 1, comma 0).

Discuss with your team how you can best approximate the area of each pie-shaped piece. Calculate the approximate area of each pie-shaped piece.
In a later problem, you will determine the exact area of this limaçon. However, at this point, what is the approximate area of the region bounded by the limaçon?

11-3.

Jessica and Ana are discussing how they determined the area of the limaçon in problem 11-2. Jessica says, “Dividing the region into slices reminds me of using rectangular slices to approximate the area under a curve. I bet that we can calculate the exact area of the region by using an integral.”

“Be careful not to cut any corners!” responds Ana. “We’re using sectors to approximate the area, not rectangles. Don’t we need a new way to calculate the area of the slices?”

Explain to Ana and Jessica why the area of a slice is approximately $A=\frac{1}{2}(f(θ))^2Δθ$ .

11-4.

Create an integral expression that represents the area of the shaded region below. Be sure to focus on both the integrand and the bounds of integration.

First quadrant curve, coming through y axis, curving up & right, & then down & left, passing through x axis, labeled, r = f of theta, with 1 shaded sector, lower arc endpoint, on the curve, labeled, theta = a, highest endpoint on the curve labeled, theta = b.

11-5.

REALITY CHECK

What is the polar equation for the circle centered at the pole (origin) with a radius of $5$ units?
Using your equation from part (a), write and evaluate an integral expression in polar form to calculate the area of this circle. Is your answer what you expected?

11-6.

Use the fact that $\cos(2θ)=2\cos^2(θ)-1$ to evaluate each of the following integrals. Homework Help ✎

$\int _ { 0 } ^ { 2 \pi } \operatorname { cos } ^ { 2 } ( \theta ) d \theta$

$\int _ { 0 } ^ { 2 \pi } 4 \operatorname { sin } ^ { 2 } ( \theta ) d \theta$

11-7.

Without a calculator, determine all values of $θ$ for which each of the polar curves cross the pole (origin). Then, graph the polar curve on your calculator and verify your answers. Homework Help ✎

$r=\ \sin(3θ)+\ \cos(3θ)$

$r=3\sin2(θ)-2$

Compute without a calculator

11-8.

If $\frac { d y } { d x }= 1 - x$ and $(–2, –4)$ is a point on the solution curve, use Euler’s Method to sketch $y$ over $–2 ≤ x ≤ 2$ . Use $Δx = 0.5$ . Homework Help ✎

11-9.

Consider the infinite series below. For each series, decide if it converges conditionally, converges absolutely, or diverges and justify your conclusion. State the tests you used. Homework Help ✎

$\displaystyle\sum _ { n = 1 } ^ { \infty } \frac { 3 } { 5 ^ { n } }$

$\displaystyle\sum _ { n = 1 } ^ { \infty } \frac { n ! } { 4 ^ { n } }$

$\displaystyle\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n + 1 } } { 3 n - 7 }$

$\displaystyle\sum _ { n = 1 } ^ { \infty } \frac { 1 } { 2 } n ^ { - 1 }$

11-10.

Let $\mathbf{a}= ⟨–2, 5⟩$ , $\mathbf{b} = ⟨6, 0⟩$ , and $\mathbf{c} =\langle \frac { 3 } { 2 } , 8 \rangle$ . What are: Homework Help ✎

$2\mathbf{c} - \mathbf{a}$

$-\mathbf{b} + \mathbf{c}$

$| | \mathbf{a} + \mathbf{b} | |$

$\frac { \mathbf{b} } { \| \mathbf{b} \| }$

11-11.

Determine all points of intersection for: Homework Help ✎
$\left\{ \begin{array} { l } { x ( t ) = 2 t } \\ { y ( t ) = t ^ { 2 } } \end{array} \right.$ and $\left\{ \begin{array} { l } { x ( t ) = t ^ { 2 } } \\ { y ( t ) = 2 t } \end{array} \right.$

11-12.

For $f(x) = \sin(x^2)$ , determine all of the relative maxima and minima over the interval $[0, 3]$ . Homework Help ✎

11-13.

Multiple Choice: As a rumor spreads across a city, its rate is proportional to the product of the number of citizens who have heard the rumor and the number who have not. Which equation below could represent this scenario? Homework Help ✎

$\frac { d P } { d t }= kP(350,000 - P)$

$\frac { d P } { d x }= kx(350,000 - x)$

$\frac { d P } { d t }= kt(350,000 - t)$

$\frac { d P } { P }= k(350,000 - t)dt$

None of these

11-14.

Multiple Choice: If $a > 0$ , $f(x) = x + ax^{–2}$ will have a relative minimum at: Homework Help ✎

$x =\sqrt { 2 a }$

$x =\sqrt [ 3 ] { 2 a }$

$x = a$

$x =\sqrt { 2 }$

$x = ± 2a$

Calculus Third Edition