Solid that lies above the square (in the xy-plane)

In summary, the conversation discusses estimating the volume of a solid that lies above a square and below an elliptic paraboloid. The volume is estimated by dividing the square into 9 equal squares and choosing sample points at the midpoints of each square. The number of squares chosen is a trade-off between accuracy and work required, with 9 being a good optimum.
  • #1
Whatupdoc
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Consider the solid that lies above the square (in the xy-plane) R= [0,1] X [01]
and below the elliptic paraboloid z= 64 -x^2 +4xy -4y^2

Estimate the volume by dividing R into 9 equal squares and choosing the sample points to lie in the midpoints of each square.


i'm not sure how you would dividing R into equal squares, cause it's an odd number. can someone help me get this started
 
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  • #2
That would be a 3X3 array of squares which is reasonable considering that the paraboloid is off-center with respect to the square. Obviously, if you divided the original square into a much larger number of areas you would get more accuracy at the expense of doing a lot more work. Fewer squares mean less work but also reduced accuracy. 9 seems like a good optimum!
 
  • #3
i don't know what i was thinking, thank you
 

1. What is a "solid that lies above the square" in the xy-plane?

A solid that lies above the square in the xy-plane refers to a three-dimensional object that extends vertically from a flat square shape on the x and y axes. This means that the object has a base that is a square and its height extends upwards from the square in the z direction.

2. How is the solid above the square different from a regular square in the xy-plane?

The solid above the square is different from a regular square in the xy-plane because the solid has an additional dimension, the z-axis, which allows it to occupy space in three dimensions. The regular square is limited to only the x and y axes, making it a two-dimensional figure.

3. Can you provide an example of a solid that lies above the square in the xy-plane?

One example of a solid that lies above the square in the xy-plane is a cube. The cube has a base that is a square and its height extends upwards, creating a three-dimensional object. Another example is a rectangular prism, which also has a base in the shape of a square and extends upwards in the z direction.

4. What are some real-world applications of solids that lie above the square in the xy-plane?

Solids that lie above the square in the xy-plane are commonly used in architecture and construction. Buildings, bridges, and other structures often have a rectangular base and extend upwards, creating a solid that lies above the square in the xy-plane. These solids are also used in geometry and engineering to represent and calculate three-dimensional objects.

5. How are solids that lie above the square in the xy-plane relevant to scientific research?

Solids that lie above the square in the xy-plane are relevant to scientific research in fields such as physics and chemistry. These solids are used to model and study the behavior of three-dimensional objects in various environments. They are also important in computer-aided design and simulations for scientific experiments and projects.

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