Using polar coordinates to find the volume of a bounded solid

In summary, the conversation discusses the use of polar coordinates to find the volume of a bounded solid. The equation of the boundary circle is found by setting z to 4 in the paraboloid and then working to get polar coordinates. The integral is set up and after double integration, the result is pi/2. There is some discussion about the integrand and a suggestion to draw a sketch to help visualize the volume being integrated.
  • #1
paraboloid
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Using polar coordinates to find the volume of a bounded solid[Solved]

zoiy4o.png

I found the equation of the boundary circle by setting z to 4 in the paraboloid.
Then I did some work to get polar coords:
[tex]x^2+y^2 = 1[/tex]
[tex]x^2+y^2 = r^2[/tex]
[tex]1-x^2-y^2 = 1-r^2[/tex]
Then I set up my integral as such
[tex]\int_0^{2\pi}\int_{0}^{1}(1-r^2)rdrd\theta[/tex]
After the double integration, I get pi/2.

edit: It should be 4r-4r3 as the integrand.
 
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  • #2
Why are you plugging 1-r2 in as the integrand? What are zupper and zlower?
 
  • #3
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  • #4
Draw a sketch of the volume you're integrating. It'll help you visualize what goes where.
 

1. What are polar coordinates?

Polar coordinates are a two-dimensional coordinate system that uses a distance from the origin and an angle from a fixed reference direction to locate a point.

2. How do you find the volume of a bounded solid using polar coordinates?

To find the volume of a bounded solid using polar coordinates, you can use the formula V = ∫∫r dr dθ, where r is the distance from the origin to the point on the solid and θ is the angle from the fixed reference direction.

3. What is the difference between using polar coordinates and Cartesian coordinates to find volume?

The main difference between using polar coordinates and Cartesian coordinates to find volume is that polar coordinates are better suited for calculating the volume of shapes with rotational symmetry, such as cylinders and cones.

4. Can polar coordinates be used to find the volume of any bounded solid?

No, polar coordinates can only be used to find the volume of bounded solids that have a rotational symmetry, such as cylinders, cones, and spheres. For other shapes, it may be more appropriate to use Cartesian coordinates.

5. What are some real-life applications of using polar coordinates to find volume?

Polar coordinates are commonly used in fields such as physics and engineering to calculate the volume of objects with rotational symmetry, such as gears, turbines, and satellite dishes. They are also used in navigation and mapping to determine the position of an object in relation to a fixed reference point.

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