Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![**Problem Statement:**
Find the area enclosed by one leaf of the rose given by the polar equation:
\[ r = 5 \cos(10\theta) \]
**Explanation:**
This equation describes a polar rose, a type of mathematical curve known for its petal-like structures. The equation \( r = 5 \cos(10\theta) \) indicates a rose curve with multiple petals. Specifically, the number of petals is determined by the coefficient of \(\theta\) in the cosine function. In this case, there are 20 petals because the coefficient is 10 (the number of petals is doubled for cosine).
**Graphical Interpretation:**
- The curve is symmetrical and rotates around the origin, creating several petals.
- Each petal is formed as \(\theta\) varies from \(0\) to \(\pi\).
- The maximum length from the origin is 5, which is the amplitude of the cosine function.
**Method for Finding Area:**
To find the area of one leaf, use the polar area formula:
\[ A = \frac{1}{2} \int_{\alpha}^{\beta} r^2 \, d\theta \]
For one petal, determine the limits of integration \(\alpha\) and \(\beta\) by solving for intervals where \(\cos(10\theta) = 0\) to find where each petal starts and ends. Here, a single petal will span an angle of \(\frac{\pi}{10}\) since the function completes its pattern over \(2\pi\).
Thus, calculate the definite integral for \(r = 5\cos(10\theta)\) over the interval corresponding to one petal.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F12c62ea9-2423-4a35-a6cd-74646c6bbd41%2F7edf38da-30fa-49cc-ac32-1bc21e1ca587%2Fgfsyxhd_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the area enclosed by one leaf of the rose given by the polar equation:
\[ r = 5 \cos(10\theta) \]
**Explanation:**
This equation describes a polar rose, a type of mathematical curve known for its petal-like structures. The equation \( r = 5 \cos(10\theta) \) indicates a rose curve with multiple petals. Specifically, the number of petals is determined by the coefficient of \(\theta\) in the cosine function. In this case, there are 20 petals because the coefficient is 10 (the number of petals is doubled for cosine).
**Graphical Interpretation:**
- The curve is symmetrical and rotates around the origin, creating several petals.
- Each petal is formed as \(\theta\) varies from \(0\) to \(\pi\).
- The maximum length from the origin is 5, which is the amplitude of the cosine function.
**Method for Finding Area:**
To find the area of one leaf, use the polar area formula:
\[ A = \frac{1}{2} \int_{\alpha}^{\beta} r^2 \, d\theta \]
For one petal, determine the limits of integration \(\alpha\) and \(\beta\) by solving for intervals where \(\cos(10\theta) = 0\) to find where each petal starts and ends. Here, a single petal will span an angle of \(\frac{\pi}{10}\) since the function completes its pattern over \(2\pi\).
Thus, calculate the definite integral for \(r = 5\cos(10\theta)\) over the interval corresponding to one petal.
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