Multiple integrals for finding volume

In summary, the conversation discusses two integration problems involving finding volumes bounded by cylindrical surfaces. The first problem involves a cylinder and two plane surfaces, while the second problem involves two cylinders. The expert suggests using horizontal or vertical slices to make the integration process easier.
  • #1
aruwin
208
0
How do I solve this? How do I determine the range? Ill they be triple integrals?Please explain to me.

Find the volumes in R3.

1. Find the volume U that is bounded by the cylinder surface x^2+y^2=1 and the plane
surfaces z=2, x+z=1.

2. Find the volume W that is bounded by the cylindrical surfaces
x^2 + y^2 = 1 and x^2 + z^2 = 1.

I know these are 2 questions and they're both about finding volumes but the second one seems to have 2 cylinders :confused: ??I don't get it.
 
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  • #2
hi aruwin! :smile:

these integration problems are basically about slicing the volume in a convenient way

for the first one …
aruwin said:
1. Find the volume U that is bounded by the cylinder surface x^2+y^2=1 and the plane
surfaces z=2, x+z=1.

… horizontal slices seem sensible:

what is the shape of the horizontal slice at a general height z ?
2. Find the volume W that is bounded by the cylindrical surfaces
x^2 + y^2 = 1 and x^2 + z^2 = 1.

I know these are 2 questions and they're both about finding volumes but the second one seems to have 2 cylinders :confused: ??I don't get it.

yes, two cylinders

again, try horizontal slices, write the second condition as x2 = 1 - z2, and remember the RHS is a constant (for that slice) :wink:
 
  • #3
tiny-tim said:
hi aruwin! :smile:

these integration problems are basically about slicing the volume in a convenient way

for the first one …


… horizontal slices seem sensible:

what is the shape of the horizontal slice at a general height z ?


yes, two cylinders

again, try horizontal slices, write the second condition as x2 = 1 - z2, and remember the RHS is a constant (for that slice) :wink:

Horizontal slice?You mean, like a triangle?? And what do you mean by RHS is a constant?:confused:
 
  • #4
aruwin said:
Horizontal slice?You mean, like a triangle??

won't one side be curved? :confused:
And what do you mean by RHS is a constant?:confused:

in x2 = 1 - z2, z is constant for any particular slice
 
  • #5
tiny-tim said:
won't one side be curved? :confused:


in x2 = 1 - z2, z is constant for any particular slice

LOL,of course one side should be a curve. My bad, I meant horizontal slice as in a slanting plane. Is that what you refer to as horizontal slice?
 
  • #6
aruwin said:
LOL,of course one side should be a curve. My bad, I meant horizontal slice as in a slanting plane. Is that what you refer to as horizontal slice?

no, a horizontal slice means between the horizontal planes at heights z and z+dz
 
  • #7
tiny-tim said:
hi aruwin! :smile:

these integration problems are basically about slicing the volume in a convenient way

for the first one …


… horizontal slices seem sensible:

what is the shape of the horizontal slice at a general height z ?


yes, two cylinders

again, try horizontal slices, write the second condition as x2 = 1 - z2, and remember the RHS is a constant (for that slice) :wink:

Can you show me a diagram for the first cylinder?I am guessing it looks like a cylinder with the one slanting surface but I can't picture the XYZ coordinate inside it. I really need to see a picture.
 
  • #8
aruwin said:
I am guessing it looks like a cylinder with the one slanting surface but I can't picture the XYZ coordinate inside it.

yes, it is a cylinder with one slanting surface :smile:

but the easiest way of integrating is to divide it into horizontal slices (so each slice is a "damaged" circle with a bit sliced off)

you could integrate by dividing it into slanting slices, but

i] you'd have to define a new parameter x+z, and integrate with respect to that

ii] you'd need to know the formula for the area of an ellipse (ok, i admit that's fairly easy)

iii] you'd still have a problem with "damaged" circles at the bottom!​
 
  • #9
1. Find the volume U that is bounded by the cylinder surface x^2+y^2=1 and the plane surfaces z=2, x+z=1.
Before leaping into calculus, think what this shape looks like. What shape might you make with two of them?

tiny-tim said:
two cylinders

again, try horizontal slices, write the second condition as x2 = 1 - z2, and remember the RHS is a constant (for that slice) :wink:

I think this one would be much easier using vertical slices in the x-z plane. The slices would be squares.
 
  • #10
haruspex said:
I think this one would be much easier using vertical slices in the x-z plane. The slices would be squares.

oooh yes … horizontal slices work, but vertical slices are easier! :smile:
 

Q1: What is a multiple integral?

A multiple integral is a mathematical concept used to find the volume under a curved surface in three-dimensional space. It involves integrating a function over a certain region, which can be thought of as finding the sum of infinitely many tiny volumes within that region.

Q2: How is a multiple integral different from a single integral?

A single integral is used to find the area under a curve in two-dimensional space, while a multiple integral is used to find the volume under a curved surface in three-dimensional space. In a multiple integral, there are multiple variables and limits, whereas in a single integral, there is only one variable and limit.

Q3: Why is a multiple integral important in science?

Multiple integrals are used in various fields of science, such as physics, engineering, and economics, to calculate important quantities such as mass, center of mass, and work. They also have applications in probability and statistics, such as finding the probability of an event occurring in a three-dimensional space.

Q4: What is the process for finding the volume using multiple integrals?

The process for finding the volume using multiple integrals involves first setting up the limits of integration, then finding the appropriate integrand (function to be integrated), and finally solving the integral using appropriate integration techniques. The result of the integral is the volume of the region bounded by the given surface.

Q5: Are there any limitations to using multiple integrals to find volume?

Yes, there are limitations to using multiple integrals to find volume. The surface must be well-defined, continuous, and have a finite volume. Additionally, the region of integration must be bounded and the integrand must be integrable over the given region. In some cases, it may be difficult or impossible to set up the integral in a way that can be solved analytically.

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