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Set up a double integral in polar coordinates to integrate the function z = f(x, y) = y over the piece of an annulus depicted below. 4- -4 -3 -2 -1 2 3 4 -2 3.

Set up a double integral in polar coordinates to integrate the function z = f(x, y) = y over the piece of an annulus depicted below. 4- -4 -3 -2 -1 2 3 4 -2 3.

Calculus: Early Transcendentals
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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Set up a double integral in polar coordinates to integrate the function \( z = f(x, y) = y^2 \) over the piece of an annulus depicted below. ### Diagram Explanation The diagram shows a section of an annulus (ring-shaped object) plotted on a grid. The annulus section is bounded by a circle centered at the origin, with an inner radius and an outer radius. The portion is a quarter circle located in the first quadrant, visually marked with a blue line. The boundaries can be deduced by visual inspection of the grid lines. ### Integral Setup The task is to express the double integral in polar coordinates \( (r, \theta) \) for the function \( y^2 \), which, in polar form, can be expressed as \( (r \sin \theta)^2 \). **Double Integral Formulation:** \[ \int \int (r \sin \theta)^2 \, r \, dr \, d\theta \] **Limits of Integration:** - **\( r \)-limits**: From the inner radius to the outer radius of the annulus. - **\( \theta \)-limits**: From the lower angle to the upper angle in radians that describe the sector of the annulus shown in the graph. These would be \(\theta = 0\) to \(\theta = \frac{\pi}{2}\) for the quarter circle. \[ \int_{0}^{\frac{\pi}{2}} \int_{r_{\text{inner}}}^{r_{\text{outer}}} (r \sin \theta)^2 \, r \, dr \, d\theta \] Fill in the specific values for the inner and outer radii as derived from the graph above.
Transcribed Image Text:Set up a double integral in polar coordinates to integrate the function \( z = f(x, y) = y^2 \) over the piece of an annulus depicted below. ### Diagram Explanation The diagram shows a section of an annulus (ring-shaped object) plotted on a grid. The annulus section is bounded by a circle centered at the origin, with an inner radius and an outer radius. The portion is a quarter circle located in the first quadrant, visually marked with a blue line. The boundaries can be deduced by visual inspection of the grid lines. ### Integral Setup The task is to express the double integral in polar coordinates \( (r, \theta) \) for the function \( y^2 \), which, in polar form, can be expressed as \( (r \sin \theta)^2 \). **Double Integral Formulation:** \[ \int \int (r \sin \theta)^2 \, r \, dr \, d\theta \] **Limits of Integration:** - **\( r \)-limits**: From the inner radius to the outer radius of the annulus. - **\( \theta \)-limits**: From the lower angle to the upper angle in radians that describe the sector of the annulus shown in the graph. These would be \(\theta = 0\) to \(\theta = \frac{\pi}{2}\) for the quarter circle. \[ \int_{0}^{\frac{\pi}{2}} \int_{r_{\text{inner}}}^{r_{\text{outer}}} (r \sin \theta)^2 \, r \, dr \, d\theta \] Fill in the specific values for the inner and outer radii as derived from the graph above.
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