Mass Given Density Function and Area

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Homework Help Overview

The problem involves calculating the mass of a quarter disc with a specified areal density function in the first quadrant. The density function is dependent on the coordinates x and y, and the discussion revolves around the integration process required to find the mass.

Discussion Character

  • Exploratory, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • Participants discuss the bounds for integration, with one participant initially using Cartesian coordinates and later converting to polar coordinates. There are questions about the correctness of the bounds and the area element used in polar coordinates.

Discussion Status

The discussion has seen multiple updates from the original poster, indicating attempts to correct their approach. Some participants have offered hints regarding the bounds and the area element, while others have pointed out potential errors in the calculations. There is no explicit consensus on the correct method yet.

Contextual Notes

Participants are working under the constraints of homework guidelines, which may limit the amount of direct assistance provided. The original poster has expressed confusion regarding the integration process and the resulting values, indicating a need for clarification on the setup and calculations.

Grace Pseudonym
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Homework Statement



A quarter disc of radius 3 cm lies in the first quadrant. The areal density is (1.2 g/cm3)x + (0.7 g/cm3)y. Determine the mass of this object.

Homework Equations



1_1.png


The Attempt at a Solution



For my bounds:
x: 0 to 3
y: 0 to Sqrt[3 - x^2]

2_1.png


When I took this integral I got 2.07846 + 5.87878 I, which is obviously an imaginary number. Any idea how to get a real number? I'm not sure where I'm going wrong-- I thought this was simple calculus 2.
 
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UPDATE: I have converted to polar coordinates, giving density = 1.2 r Cos[theta]+ 0.7 r Sin[theta]. Then I solved the following integral:

3_1.png


This gave me 8.55 g which was incorrect. Where am I going wrong?
 
Your bound for y is wrong, draw a diagram with your bounds (e.g. y as a function of x) and you'll see that's not the bounds of a circle quadrant. Hint: the relation is x^2 + y^2 = 3^2
 
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Emily Herschbein said:
UPDATE: I have converted to polar coordinates, giving density = 1.2 r Cos[theta]+ 0.7 r Sin[theta]. Then I solved the following integral:

3_1.png


This gave me 8.55 g which was incorrect. Where am I going wrong?
If you check the units of your expression, you'll see that they don't work out. Your mistake is in the area element for polar coordinates.
 
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UPDATE NO. 2:

I figured it out.

they.png
 
Thank you everyone for your help :)
 
You stumbled onto the correct answer, but your work isn't correct.
 

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