Triple integral in cylindrical coordinates

  • Thread starter Calpalned
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Homework Statement


Evaluate ## \int \int \int_E {x}dV ## where E is enclosed by the planes ##z=0## and ##z=x+y+5## and by the cylinders ##x^2+y^2=4## and ##x^2+y^2=9##.

Homework Equations



## \int \int \int_E {f(cos(\theta),sin(\theta),z)}dzdrd \theta ##
How do I type limits in for integration?

The Attempt at a Solution


Right now I'm just trying to find the limits of integration.

For dz, ##z=x+y+5## is equivalent to ##z=rcos \theta +r sin \theta +5## so that is the upper limit for ##z## while ##z=0## is the lower limit.

For dθ I am going to assume that it is ##0 < \theta < 2 \pi ## By the way, how do I make the "greater than or equal to" sign in Latex? I choose zero and two pi because question didn't say the object is restricted to any octant. Keep in mind that I do not know what the graph looks like visually so I am taking a risk...

For dr, I have ##r^2 = 4## and ##r^2 = 9## what do I do? Additionally, can the lower limit for r ever be negative?
 

Answers and Replies

  • #2
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Homework Statement


Evaluate ## \int \int \int_E {x}dV ## where E is enclosed by the planes ##z=0## and ##z=x+y+5## and by the cylinders ##x^2+y^2=4## and ##x^2+y^2=9##.

Homework Equations



## \int \int \int_E {f(cos(\theta),sin(\theta),z)}dzdrd \theta ##
In polar and cylindrical coordinates you're going to need an extra factor of r, as in ##r~dr~d\theta##, which is equivalent to dx dy in rectangular coordinates.
Calpalned said:
How do I type limits in for integration?
# # \int_{a}^{b} ... # #
Here a is the lower limit and b is the upper limit. If either limit is a single character, you don't need the braces. However, if a limit consists of two or more characters, like -3, then you need the braces.
Calpalned said:

The Attempt at a Solution


Right now I'm just trying to find the limits of integration.

For dz, ##z=x+y+5## is equivalent to ##z=rcos \theta +r sin \theta +5## so that is the upper limit for ##z## while ##z=0## is the lower limit.

For dθ I am going to assume that it is ##0 < \theta < 2 \pi ## By the way, how do I make the "greater than or equal to" sign in Latex?
\ge for ≥ and \le for ≤
Calpalned said:
I choose zero and two pi because question didn't say the object is restricted to any octant. Keep in mind that I do not know what the graph looks like visually so I am taking a risk...
Then you should sketch a graph of your region. It's not that complicated. The region is a hollow tube whose walls are formed by the two cylinders, with the cap being the plane z = x + y + 5. This plane makes a diagonal slice through the tube.
Calpalned said:
For dr, I have ##r^2 = 4## and ##r^2 = 9## what do I do? Additionally, can the lower limit for r ever be negative?
You can take r = 2 and r = 3. It's possible for r to be negative, due to the ambiguity of coordinates in polar and cylindrical coordinates. By that I mean that a single point can have multiple representations, which is not the case in rectangular coordinates. For example (-1, 7π/6) is the same point as (1, π/6).
 

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