Volume of solid with given base and cross sections

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Homework Help Overview

The problem involves finding the volume of a solid whose base is defined by the semicircle \( y = \sqrt{9 - x^2} \) for \( -3 \leq x \leq 3 \), with cross sections perpendicular to the x-axis being squares. Participants are exploring how to calculate the cross-sectional area and the overall volume using integration.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning, Problem interpretation

Approaches and Questions Raised

  • Participants discuss the relationship between the semicircle and the square cross sections, with one participant questioning how to express the side length of the square in terms of y. Others suggest that the problem may be approached differently, hinting at simpler methods for finding the volume.

Discussion Status

The discussion is ongoing, with participants sharing different perspectives on the complexity of the problem. Some guidance has been offered regarding the relationship between the semicircle and the area of the square, but there is no consensus on the best approach to take.

Contextual Notes

There is a mention of the problem being part of a section in a textbook focused on finding volumes through integration, which may impose certain constraints on the methods used. Additionally, there are concerns about the necessity of using advanced math for what some perceive as a simpler problem.

elsternj
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Homework Statement


The base is the semicircle y = \sqrt{9-x<sup>2</sup>}(Square root of 9-x2.. i don't know why the formatting isn't showing up) where -3 <= x <= 3. The cross section perpendicular to the x-axis are squares.



Homework Equations



-3\int3 = A(y)dy
A(y) = area of cross section

The Attempt at a Solution


Okay so i know the cross section is a square. And i see that y = \sqrt{9-x<sup>2</sup>} creates a semi-circle. I am not sure how to find the cross sectional area. I'm assuming it needs to be in terms of y. And the area of a square is a (side)2. But how do i know what the side is equal to?
 
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Looks to me to be a fairly simply solid (half a cylinder) so I think you're going about it the hard way.
 
hmm well the reason I am going about it this way is because the problem is in the section of the book where we are finding volumes by integrating the area of the cross section. I am not aware of another way to do this problem.
 
I'm not a big fan of simple problems requiring the use of more advanced math than is needed but I do understand that the math you're studying needs practice problems so I guess you'll need to do it the hard way.
 
elsternj said:

Homework Statement


The base is the semicircle y = \sqrt{9-x<sup>2</sup>}(Square root of 9-x2.. i don't know why the formatting isn't showing up) where -3 <= x <= 3. The cross section perpendicular to the x-axis are squares.

Homework Equations



-3\int3 = A(y)dy
A(y) = area of cross section

The Attempt at a Solution


Okay so i know the cross section is a square. And i see that y = \sqrt{9-x<sup>2</sup>} creates a semi-circle. I am not sure how to find the cross sectional area. I'm assuming it needs to be in terms of y. And the area of a square is a (side)2. But how do i know what the side is equal to?
You are told that the base of the square is in the semi-circle from y= 0 up to y= \sqrt{9- x^2}. That is the length of the side. The area of the square is 9- x^2.
 
but of course when you do it the easy way, the area of the square is irrelevant and the answer pretty much just pops out. When you do it the hard way you should get that the volumn is 27*pi
 

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