Integrating triple integral over region W

In summary, the author is trying to find the volume of a region bounded by a plane, a cylinder, and a vertical line. He has trouble deciding how to proceed and decides to use the function x^2+z^2=1. He runs into a problem when trying to decide the bounds for x and z. He decides to use polar coordinates instead and finds that it is okay. However, he gets a wrong answer after following @RUber 's suggestion of changing the order of integration. He finally solves the problem using Maple.
  • #1
toforfiltum
341
4

Homework Statement


$$f(x,y,z)=y$$ ; W is the region bounded by the plane ##x+y+z=2##, the cylinder ##x^2 +z^2=1##, and ##y=0##.

Homework Equations

The Attempt at a Solution


Since there is a plane of ##y=0##, I decided that my inner integral will be ##y=0## and ##y=2-x-z##. But after this I have a problem. I don't know how to proceed. How do I decide the bounds of ##x## and ##z##? Can I just use the function ##x^2+z^2=1##? If so, will it be $$\int_{-1}^1 \int_{-\sqrt(1-x^2)}^{\sqrt(1-x^2)} \int_ {0}^{2-x-z} y \,dy \, dz \, dx$$

Thanks.
 
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  • #2
toforfiltum said:

Homework Statement


$$f(x,y,z)=y$$ ; W is the region bounded by the plane ##x+y+z=2##, the cylinder ##x^2 +z^2=1##, and ##y=0##.

Homework Equations

The Attempt at a Solution


Since there is a plane of ##y=0##, I decided that my inner integral will be ##y=0## and ##y=2-x-z##. But after this I have a problem. I don't know how to proceed. How do I decide the bounds of ##x## and ##z##? Can I just use the function ##x^2+z^2=1##? If so, will it be $$\int_{-1}^1 \int_{-\sqrt(1-x^2)}^{\sqrt(1-x^2)} \int_ {0}^{2-x-z} y \,dy \, dz \, dx$$

Thanks.

Yes, you can do it that way. It might be easier to use polar type coordinates for the xz integral because of the circle, depending on how easy the rectangular integral is.
 
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  • #3
LCKurtz said:
Yes, you can do it that way. It might be easier to use polar type coordinates for the xz integral because of the circle, depending on how easy the rectangular integral is.
Yes, the rectangular integral is really horrible. It involved a lot of ##cos^4 \theta## terms.
 
  • #4
I agree with LCKurtz. Change to cylindrical coordinates. You have to account for the place where the plane starts to intersect the cylinder. Otherwise, you'll be integrating over the entire circle near the top, and won't get the right volume.
 
  • #5
RUber said:
I agree with LCKurtz. Change to cylindrical coordinates. You have to account for the place where the plane starts to intersect the cylinder. Otherwise, you'll be integrating over the entire circle near the top, and won't get the right volume.
But I think I'm supposed to be calculating it using rectangular coordinates, because this question is in the section before the change of variables topic.
 
  • #6
Okay then. Maybe look at changing the order of integration? Let y be your outermost integral, since limits on x and z will both depend on how much of the circle is being cut off by the plane.
It looks like there are two parts here...the first part is just the cylinder before the plane intersects it.
The second part will be from y = 1 to y = 3. Try to define your x and z limits based on that.
 
  • #7
I'm not sure I understand @RUber 's concerns and suggestions. I would say you definitely want to integrate in the ##y## direction first, and the remaining ##x,y## or ##r,\theta## integral will be over the whole circle in the xz plane. I think you should get ##\frac{9\pi}{4}## either way.
 
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  • #8
LCKurtz said:
I'm not sure I understand @RUber 's concerns and suggestions. I would say you definitely want to integrate in the ##y## direction first, and the remaining ##x,y## or ##r,\theta## integral will be over the whole circle in the xz plane. I think you should get ##\frac{9\pi}{4}## either way.
Ok, thanks for providing the correct answer for me to check. However, I got a wrong answer. What I would like to know is if the process of calculating the integral in rectangular coordinates really long? Because mine certainly is. I'm just wondering if I'm doing it correctly.
 
  • #9
@LCKurtz Oh, it's fine. I've got the correct answer now, though only after a painfully long process. :smile:

Thanks!
 
  • #10
toforfiltum said:
@LCKurtz Oh, it's fine. I've got the correct answer now, though only after a painfully long process. :smile:

Thanks!

Good job. Luckily, I didn't experience the "pain" of working it all out, since I just put your original correct triple integral into Maple. It cranked out the answer in a couple of seconds.
 

1. What is a triple integral?

A triple integral is a type of mathematical operation that involves integrating a function over a three-dimensional region in space. It is an extension of the concept of a single integral to three dimensions.

2. How is a triple integral over a region W calculated?

To calculate a triple integral over a region W, you first need to define the limits of integration for each of the three variables (x, y, and z) in the region. Then, you integrate the function over each variable, using the corresponding limits of integration, and multiply the results together.

3. What are some real-world applications of integrating triple integrals?

Triple integrals have many applications in physics, engineering, and other fields. They can be used to calculate the volume of a three-dimensional object, the mass of a solid with varying density, the center of mass of an object, and the moment of inertia of an object, among others.

4. What are some common techniques for solving triple integrals?

Some common techniques for solving triple integrals include using symmetry to simplify the integrand, using change of variables, and using cylindrical or spherical coordinates instead of Cartesian coordinates. It is also important to carefully consider the order of integration in order to make the calculations more manageable.

5. What is the importance of integrating triple integrals in scientific research?

Integrating triple integrals is an important tool in scientific research as it allows for the analysis and calculation of complex quantities in three-dimensional space. It is also used in developing mathematical models for physical systems and in solving problems in fields such as physics, engineering, and economics.

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