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...
Related questions
Question
f(r,theta)= 4(rcos(theta)+rsin(theta)
region over r is in between 1 and 5
region over theta is inbetween 0 and pi/3
![## Double Integral in Polar Coordinates
Thus, the double integral of a function \( f(x, y) \) over a region \( A \) is given by:
\[
\iint f(x, y) \, dA = \int_{0}^{\frac{\pi}{3}} \int_{1}^{5} \boxed{\quad} \, r \, dr \, d\theta
\]
### Explanation:
- **Outer Integral**: \(\int_{0}^{\frac{\pi}{3}} d\theta\) — This integral represents the integration over the angular range from \( 0 \) to \( \frac{\pi}{3} \).
- **Inner Integral**: \(\int_{1}^{5} r \, dr\) — This part represents the integration over the radial range from \( 1 \) to \( 5 \), multiplied by the factor \( r \), which accounts for the transformation from Cartesian to polar coordinates. The placeholder box indicates that the integrand should be specified.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F09d105f3-6c69-4cbc-8997-6988f1733e6f%2F6618481c-568f-4493-b144-8900df119f66%2Fkni7l8b_processed.png&w=3840&q=75)
Transcribed Image Text:## Double Integral in Polar Coordinates
Thus, the double integral of a function \( f(x, y) \) over a region \( A \) is given by:
\[
\iint f(x, y) \, dA = \int_{0}^{\frac{\pi}{3}} \int_{1}^{5} \boxed{\quad} \, r \, dr \, d\theta
\]
### Explanation:
- **Outer Integral**: \(\int_{0}^{\frac{\pi}{3}} d\theta\) — This integral represents the integration over the angular range from \( 0 \) to \( \frac{\pi}{3} \).
- **Inner Integral**: \(\int_{1}^{5} r \, dr\) — This part represents the integration over the radial range from \( 1 \) to \( 5 \), multiplied by the factor \( r \), which accounts for the transformation from Cartesian to polar coordinates. The placeholder box indicates that the integrand should be specified.
Expert Solution
Step 1
Given that the function .
Find the value of given integral.
The formula used are the following.
where is a constant.
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