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Ch.12 - Parametric and Polar Curves
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh.12 - Parametric and Polar CurvesProblem 12.3.80a
Chapter 12, Problem 12.3.80a

Area of roses Assume m is a positive integer.


a. Even number of leaves: What is the relationship between the total area enclosed by the 4m-leaf rose r=cos(2mθ) and m?

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Recall that the polar equation of a rose curve is given by \(r = \cos(k\theta)\), where \(k\) determines the number of petals (or leaves) of the rose. For \(k = 2m\), the rose has \$4m\( leaves because when \)k\( is even, the number of petals is \)2k$.
The area of one petal of a rose curve \(r = \cos(k\theta)\) can be found using the formula for the area in polar coordinates: \(A = \frac{1}{2} \int_{\alpha}^{\beta} r^2 \, d\theta\). Here, \(r^2 = \cos^2(k\theta)\).
To find the total area enclosed by the \$4m\(-leaf rose, calculate the area of one petal and then multiply by the total number of petals, which is \)4m$.
Determine the limits of integration for one petal. Since the rose has \$4m$ petals evenly distributed over \(2\pi\), the angle for one petal is \(\frac{2\pi}{4m} = \frac{\pi}{2m}\). So, integrate \(\theta\) from \(0\) to \(\frac{\pi}{2m}\) for one petal.
Set up the integral for the area of one petal: \(A_{petal} = \frac{1}{2} \int_0^{\frac{\pi}{2m}} \cos^2(2m\theta) \, d\theta\). Then multiply by \$4m$ to get the total area: \(A_{total} = 4m \times A_{petal}\). Use trigonometric identities to simplify the integral if needed.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Polar Coordinates and Graphing

Polar coordinates represent points using a radius and angle, ideal for curves like roses. The equation r = cos(kθ) produces rose curves with petals depending on k. Understanding how to plot and interpret these curves is essential for analyzing their geometric properties.
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Intro to Polar Coordinates

Rose Curves and Petal Count

Rose curves defined by r = cos(kθ) have petals determined by k: if k is even, the curve has 2k petals; if odd, k petals. For r = cos(2mθ), with m positive integer, the curve has 4m petals. Recognizing this helps relate the number of petals to the parameter m.
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Roses

Area Calculation in Polar Coordinates

The area enclosed by a polar curve r(θ) from θ = a to b is given by (1/2)∫[a to b] (r(θ))^2 dθ. For rose curves, integrating over one petal and multiplying by the number of petals yields total area. This integral approach is key to finding the relationship between area and m.
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Intro to Polar Coordinates
Related Practice
Textbook Question

11–14. Working with parametric equations Consider the following parametric equations.

a. Make a brief table of values of t, x, and y.

b. Plot the (x, y) pairs in the table and the complete parametric curve, indicating the positive orientation (the direction of increasing t).


x=−t+6, y=3t−3; −5≤t≤5 

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Textbook Question

11–14. Working with parametric equations Consider the following parametric equations.

a. Make a brief table of values of t, x, and y.

b. Plot the (x, y) pairs in the table and the complete parametric curve, indicating the positive orientation (the direction of increasing t).


x=2 t,y=3t−4;−10≤d≤10 

Textbook Question

The ellipse and the parabola: Let R be the region bounded by the upper half of the ellipse x²/2 + y² = 1 and the parabola y = x²/√2

a. Find the area of R

Textbook Question

Reflection property of parabolas: Consider the parabola y = x²/(4p) with its focus at F(0, p). The goal is to show that the angle of incidence (α) equals the angle of reflection (β).

a. Let P(x₀, y₀) be a point on the parabola. Show that the slope of the tangent line at P is tan θ = x₀/(2p).

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Textbook Question

Area of a sector of a hyperbola: Consider the region R bounded by the right branch of the hyperbola x²/a² - y²/b² = 1 and the vertical line through the right focus

a. What is the area of R?

Textbook Question

Navigating A plane is 150 miles north of a radar station, and 30 minutes later it is 60 degree east of north at a distance of 100 miles from the radar station. Assume the plane flies on a straight line and maintains constant altitude during this 30-minute period.

a. Find the distance traveled during this 30-minute period.

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