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

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In summary, the conversation is discussing three different problems that involve finding the volume of a solid and evaluating an integral using a change of variables. The first problem involves finding the volume of a solid bounded by two cylinders on different planes. The second problem involves finding the volume and centroid of a region bounded by a sphere and a cone. The third problem involves using a change of variables to evaluate an integral for a region bounded by the x-axis and two parabolas. The person is seeking help for all three problems and is unsure of how to approach them.
  • #1
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.
 
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  • #3
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.
 

1. What is the volume of a region?

The volume of a region is the amount of 3-dimensional space that it occupies.

2. How do you calculate the volume of a region using integrals?

To calculate the volume of a region using integrals, you need to first find the cross-sectional area of the region at each point along the axis of rotation. Then, integrate the cross-sectional areas over the desired interval to find the total volume.

3. What is the centroid of a region?

The centroid of a region is the geometric center or balance point of the region, where it would balance perfectly if it were placed on a fulcrum.

4. How do you find the centroid of a region?

To find the centroid of a region, you can use the formula x̄ = (1/A) ∫ xy dA for the x-coordinate and ȳ = (1/A) ∫ y dA for the y-coordinate, where A is the area of the region and the integrals are taken over the region.

5. What is the relationship between the volume and centroid of a region?

The centroid of a region can be thought of as the balance point of the region, and the volume can be thought of as the distribution of weight in the region. Therefore, the centroid is the point where the weight is evenly distributed and the volume is evenly distributed on all sides.

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