Is this double integral set up correctly?

In summary, the problem given does not have a solution as the three given equations do not bound a region of finite volume.
  • #1
G01
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1.Set up the integral to Find the volume enclosed by the cylinder x^2 +y^2 = 1 x=0 and z=y


3. The area to integrate over is the part of x^2 + y^2 =1 above the x axis. X goes from -1 to 1 and y goes from 0 to sqrt(1-x^2) So the integral should be:

[tex] \int^1_{-1} \int^{\sqrt(1-x^2)}_0 y dy dx [/tex]

 
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  • #2
The way you phrased the question, I don't think that the volume is above the x-axis (or the x-plane). The x=0 plane is the yz plane, so it is actually cutting your cylinder in-half vertically and the the z=y plane is slashing your cylinder diagonally.
 
  • #3
G01 said:
1.Set up the integral to Find the volume enclosed by the cylinder x^2 +y^2 = 1 x=0 and z=y


3. The area to integrate over is the part of x^2 + y^2 =1 above the x axis. X goes from -1 to 1 and y goes from 0 to sqrt(1-x^2) So the integral should be:

[tex] \int^1_{-1} \int^{\sqrt(1-x^2)}_0 y dy dx [/tex]

Please go back and recheck the statement of the problem. x2+ y2[/suo] is an infinite cylinder and x= 0 is a plane crossing the x-axis. Together they bound an infinite "half-cylinder". z= y is a plane that crossing that half-cylinder. The three together do NOT bound a region of finite volume. The region either above or below z= y does not have finite volume. If the second equation were z= 0, then it would make sense.
 

1. What is a double integral?

A double integral is a mathematical concept that involves evaluating the area under a two-dimensional function over a specific region on a coordinate plane.

2. How do you set up a double integral?

The set up of a double integral depends on the type of region being integrated over. Generally, the inner integral is set up for the x-axis and the outer integral is set up for the y-axis. The limits of integration are determined by the boundaries of the region.

3. What is the purpose of a double integral?

A double integral is used to calculate the total value of a two-dimensional function over a specific region. It is commonly used in physics, engineering, and other scientific fields to calculate quantities like mass, volume, and moment of inertia.

4. What are some common mistakes in setting up a double integral?

One common mistake is mixing up the order of the integrals or incorrectly setting the limits of integration. It is important to carefully consider the boundaries of the region and the orientation of the integrals to avoid these mistakes.

5. How do you know if a double integral is set up correctly?

A double integral is set up correctly if the limits of integration correspond to the boundaries of the region and the integrals are correctly ordered. Additionally, the final answer should make sense in the context of the problem being solved.

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