Finding volume in Polar Coordinates

In summary, the problem asks for the volume of a wedge-shaped region contained in a cylinder with the equation x^2+y^2=9, bounded by the plane z=x and below by the xy plane. The solution involves setting up a triple integral in polar coordinates, with the limits of integration as follows: dr between 0 and 3, dz between 0 and rcosθ, and dθ between 0 and 2∏. It is important to visualize the region and take into account the bound on z and theta. The range of possible values for theta can be determined by considering the constraints given.
  • #1
PsychonautQQ
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Homework Statement


Find the volume of the wedge-shaped region contained in the cylinder x^2+y^2=9 bounded by the plane z=x and below by the xy plane

Homework Equations


The Attempt at a Solution


So it seems a common theme for me I have a hard time finding the limits of integration for the dθ term when I integrate in polar coordinates.

For this integral I am setting it up
triple integral rdzdrdθ where dr is between 0 and 3, dz is between 0 and rcosθ and dθ is between 0 and 2∏? is that correct?
 
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  • #2
You need to be careful with this one. There's no substitute (AFAIK) for visualising the region. Pay particular attention to the fact that it says below by the XY plane. That imposes a bound on z, and hence on theta.
 
  • #3
does theta go from 0 to 45 since z=x it will create a 45 degree angle? dr between 0 and 3, and dz between 0 and rcos(theta)?
 
  • #4
Show us a picture of what you have for the region.
 
  • #5
PsychonautQQ said:
does theta go from 0 to 45 since z=x it will create a 45 degree angle? dr between 0 and 3, and dz between 0 and rcos(theta)?
Theta is an angle in the XY plane, so is not directly related to z = x.
If 0 <= z <= x and x = r cos θ and r > 0, what is the range of possible values for theta?
 
  • #6
Why should one use cylinder coordinates here to begin with?
 

1. What is the formula for finding volume in polar coordinates?

The formula for finding volume in polar coordinates is V = ∫∫∫ r sin(θ) dr dθ dz, where r represents the radius, θ represents the angle, and z represents the height.

2. How do you convert Cartesian coordinates to polar coordinates?

To convert Cartesian coordinates (x,y,z) to polar coordinates (r,θ,z), you can use the equations r = √(x^2 + y^2), θ = tan^-1(y/x), and z = z. These equations can be derived from the Pythagorean theorem and trigonometric functions.

3. Can you find volume in polar coordinates for irregular shapes?

Yes, it is possible to find volume in polar coordinates for irregular shapes. This can be done by breaking down the shape into smaller, simpler shapes and using the formula V = ∫∫∫ r sin(θ) dr dθ dz to find the volume for each individual shape. The volumes can then be added together to get the total volume of the irregular shape.

4. What are the limitations of using polar coordinates to find volume?

One limitation of using polar coordinates to find volume is that it is not as straightforward as using Cartesian coordinates for simpler shapes such as cubes or spheres. Additionally, polar coordinates may not be the most efficient method for finding volume for certain shapes, such as those with varying cross-sectional areas.

5. Are there any real-world applications of finding volume in polar coordinates?

Yes, there are several real-world applications of finding volume in polar coordinates. This includes calculating the volume of objects such as cones, cylinders, and spheres, as well as finding the volume of fluids in circular containers or pipes. It can also be used in physics and engineering for calculating the volume of rotating objects or calculating the mass of objects with varying densities.

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