Help Needed: Volume & Centroid of Region, Integral of Parabolas

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Homework Help Overview

The discussion revolves around finding the volume and centroid of regions defined by various geometric shapes, including cylinders and spheres, as well as evaluating integrals using change of variables. The subject area includes multivariable calculus and geometric interpretations of integrals.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • The original poster attempts to understand the setup for three distinct problems involving volumes and centroids, expressing confusion about bounds and initial steps. Some participants suggest fixing a value for y to simplify the first problem, while others question the connection to another thread.

Discussion Status

The discussion is ongoing, with participants exploring different approaches to the problems. Some guidance has been offered regarding the first problem, but there is no consensus on how to proceed with the remaining questions. The original poster continues to seek clarity on the setup and methods.

Contextual Notes

The original poster indicates a lack of progress after two days of work on the problems, suggesting potential constraints in understanding or resources. There is also a reference to another thread, which may imply overlapping homework topics.

LUmath09
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I have been looking at these 3 problems for 2 days now and have gotten practically nowhere. Help please!

1. Find the volume V of the solid bounded by the graph of x2 + y2 = 9 and y2 + z2 = 9.

I know that both equations are cylinders on different planes and that I need the intersection. I can not figure out what my bounds are or how to set up the problem. I'm stuck.


2. Find the volume and the centroid (center of mass) of the region that is bounded above by the sphere ρ = a and below by the cone φ = c with 0 < c < π/2. Here you assume constant density.


3. Use the change of variables x = u2 - v2, y = 2uv to evaluate the ∫∫R ydA , where R is the region bounded by the x - axis and the parabolas y2 = 4 − 4x and y2 = 4 + 4x.


I have no idea how to even begin the last two. I've looked through the book and through my notes and can not come up with anything.
 
Last edited:
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One at a time, ok? Fix a value of y. Then x^2=9-y^2 and z^2=9-y^2. So both x and z range from -sqrt(9-y^2) to +sqrt(9-y^2). I.e. for fixed y the x-z cross-section is a SQUARE.
 

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