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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Q4 need help please explain
![### Explanation of the Image
The image shows a mathematical problem related to polar coordinates. It presents the polar curve given by \( r = 5 \sin(4\theta) \) and displays one of its branches on a graph.
#### Graph Description:
- The curve is depicted in blue on a two-dimensional plane with horizontal (x-axis) and vertical (y-axis) lines representing the coordinate axes.
- The shape is a petal-like figure extending from the origin, typical for polar equations of the form \( r = a \sin(k\theta) \) where \( k \) determines the number of petals.
- In this particular graph, one section of the curve is visible, which is part of the complete set typically shown in polar plots.
#### Integral Setup:
The task involves determining the area enclosed by the curve using integration. Below the graph, the text mentions:
"An integral to find the area of the region is"
It is followed by a blank integral expression \( \int \) with empty boxes for input, accompanied by \( d\theta \). The instructions specify to "Type the word 'theta' for \( \theta \)".
### Educational Context
Students are asked to calculate the area of one leaf of the polar curve using the integral formula for polar coordinates:
\[
\text{Area} = \frac{1}{2} \int_{\alpha}^{\beta} r^2 \, d\theta
\]
where \( r \) is a function of \( \theta \) as given by the equation \( r = 5 \sin(4\theta) \). To complete the exercise, students must determine the correct limits of integration \( \alpha \) and \( \beta \) that correspond to the beginning and end of one petal of the curve.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8c5164fa-e600-4317-9bd4-8c706ea2635f%2F38f206d0-0135-412a-a2c9-1a1d9f4b2623%2Fxj9k9t_processed.png&w=3840&q=75)
Transcribed Image Text:### Explanation of the Image
The image shows a mathematical problem related to polar coordinates. It presents the polar curve given by \( r = 5 \sin(4\theta) \) and displays one of its branches on a graph.
#### Graph Description:
- The curve is depicted in blue on a two-dimensional plane with horizontal (x-axis) and vertical (y-axis) lines representing the coordinate axes.
- The shape is a petal-like figure extending from the origin, typical for polar equations of the form \( r = a \sin(k\theta) \) where \( k \) determines the number of petals.
- In this particular graph, one section of the curve is visible, which is part of the complete set typically shown in polar plots.
#### Integral Setup:
The task involves determining the area enclosed by the curve using integration. Below the graph, the text mentions:
"An integral to find the area of the region is"
It is followed by a blank integral expression \( \int \) with empty boxes for input, accompanied by \( d\theta \). The instructions specify to "Type the word 'theta' for \( \theta \)".
### Educational Context
Students are asked to calculate the area of one leaf of the polar curve using the integral formula for polar coordinates:
\[
\text{Area} = \frac{1}{2} \int_{\alpha}^{\beta} r^2 \, d\theta
\]
where \( r \) is a function of \( \theta \) as given by the equation \( r = 5 \sin(4\theta) \). To complete the exercise, students must determine the correct limits of integration \( \alpha \) and \( \beta \) that correspond to the beginning and end of one petal of the curve.
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