What is the triple integral of z^2(x^2 + y^2) over a bounded cylindrical region?

In summary: Good job!In summary, we are given a bounded cylindrical region with the equation W= {(x,y,z)| x^2 + y^2 ≤ 1, -1 ≤ z ≤ 1}. The task is to evaluate the triple integral of the function f(x,y,z)= z^2 x^2 + z^2 y^2 over W using cylindrical coordinates. After converting the function and limits to cylindrical coordinates, we integrate with the limits r ≤ 1, 0≤ theta ≤ 2pi, and -1≤z≤1 and the differential element rdrdzdtheta. The simplified function to be integrated is z^2 r^2. The final answer is pi/6.
  • #1
3soteric
23
0

Homework Statement



Let W= {(x,y,z)| x^2 + y^2 ≤ 1, -1 ≤ z ≤ 1} (W is a bounded cylindrical region)

Evaluate the triple integral f(x,y,z)= z^2 x^2 + z^2 y^2 over W. Use cylindrical coordinates

Homework Equations


i don't see any relevant equations besides the obvious cylindrical coordinate equations.


The Attempt at a Solution


I have no clue what the problem is asking for to be honest? any help on how to get me started? thank you
 
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  • #2
convert the function to cyl coordinates using x = r cos(theta) and y=r sin(theta) and z=z and what do you get?

Can you simplify it?
 
  • #3
jedishrfu said:
convert the function to cyl coordinates using x = r cos(theta) and y=r sin(theta) and z=z and what do you get?

Can you simplify it?

yes sir! I am on it right now! thanks
 
  • #4
3soteric said:

Homework Statement



Let W= {(x,y,z)| x^2 + y^2 ≤ 1, -1 ≤ z ≤ 1} (W is a bounded cylindrical region)

Evaluate the triple integral f(x,y,z)= z^2 x^2 + z^2 y^2 over W. Use cylindrical coordinates

Homework Equations


i don't see any relevant equations besides the obvious cylindrical coordinate equations.


The Attempt at a Solution


I have no clue what the problem is asking for to be honest? any help on how to get me started? thank you

Express your function and limits in term of the cylindrical coordinates. Change (x,y,z) to (r,θ,z) and don't forget to change dxdydz to its cylindrical equivalent. Then integrate. That's what it is asking you.
 
  • #5
f(x,y,z)= z^2 (r^2 cos^2theta) + z^2 (r^2 sin^2 theta)
 
  • #6
thank you dick! I am on it right now ! ill post results
 
  • #7
can i integrate the first limit then convert to polar? or do i have to flat out use cylindrical coordinates?
 
  • #8
3soteric said:
can i integrate the first limit then convert to polar? or do i have to flat out use cylindrical coordinates?

There's not that much of a difference. "cylindrical" is just "polar" plus the extra coordinate z. Use what you know about polar.
 
  • #9
if i draw the xz and yz traces, will it be a horizontal line at 1 and at -1? then of course the xy trace is a circle with r=1
 
  • #10
i got the following limits, r ≤ 1,
0≤ theta ≤ 2pi ,

-1≤z≤1

thats in polar right? now do i integrate? rdrdzdtheta?
 
  • #11
3soteric said:
if i draw the xz and yz traces, will it be a horizontal line at 1 and at -1? then of course the xy trace is a circle with r=1

The intersection with the xz and yz planes are rectangles, as the problem said, it's a cylinder. I'm not sure why you worried about this.
 
  • #12
3soteric said:
i got the following limits, r ≤ 1,
0≤ theta ≤ 2pi ,

-1≤z≤1

thats in polar right? now do i integrate? rdrdzdtheta?

Those are the limits and that's the differential element alright. So what function will you integrate?
 
  • #13
Dick said:
Those are the limits and that's the differential element alright. So what function will you integrate?

the function i will integrate will be z^2 x^2 + x^2 y^2 which i need to convert so its
z^2 (r^2 cos^2theta) + z^2 (r^2 sin^2 theta)

i integrate that? if so do i integrate rdzdrdtheta or rdrdzdtheta? i can't find a limit for dz, is it just -1 to 1? i don't have to express in other variables ? if so then its

∫2pi-0 ∫1-0 ∫ 1-(-1) (the function) r dz dr dtheta?
 
  • #14
by the way thanks for helping dick
 
  • #15
3soteric said:
by the way thanks for helping dick

No problem. But you can simplify the function you are integrating. Use a trig identity. It looks like you've got the rest ok.
 
  • #16
can i use sin^2+cos^2=1 then basically do z^2 r^2 + z^2 r^2 which is 2z^2r^2?
 
  • #17
3soteric said:
can i use sin^2+cos^2=1 then basically do z^2 r^2 + z^2 r^2 which is 2z^2r^2?

You got the identity right. But the result came out wrong. Give that another thought.
 
  • #18
i conclude z^2 (r^2 cos^2 theta + r^2 sin^2 theta) so i can factor the r^2 as well and I am left with (sin ^2 + cos ^2) and that's one

so z^2 r^2?
 
  • #19
3soteric said:
i conclude z^2 (r^2 cos^2 theta + r^2 sin^2 theta) so i can factor the r^2 as well and I am left with (sin ^2 + cos ^2) and that's one

so z^2 r^2?

Much better. Now put it all together.
 
  • #20
will do ! ill post back with results thank you
 
  • #21
my answer ended up being 4pi/9!
 
  • #22
3soteric said:
my answer ended up being 4pi/9!

Did you remember to include the r in r dz dr dtheta?
 
  • #23
Dick said:
Did you remember to include the r in r dz dr dtheta?

yep basically i did this


∫2pi-0 ∫1-0 ∫ 1-(-1) z^2 r^3 dz dr dtheta

i multiplied z^2r^2 by r so i got r^3
(first digit means upper limit for integral)
 
  • #24
op5lwl.jpg
 
  • #25
correction answer is pi/6 right?
 
  • #26
3soteric said:
correction answer is pi/6 right?

Yes, pi/6.
 

1. What are cylindrical coordinates?

Cylindrical coordinates are a type of coordinate system used in mathematics and physics to represent points in three-dimensional space. It consists of a radial distance from a central axis, an angle from a reference direction, and a height from a reference plane. This system is particularly useful for describing objects with cylindrical symmetry.

2. How do cylindrical coordinates differ from Cartesian coordinates?

Cylindrical coordinates differ from Cartesian coordinates in that they use a different set of variables to represent points in three-dimensional space. While Cartesian coordinates use x, y, and z axes, cylindrical coordinates use r, θ, and z axes. This allows for a more intuitive representation of objects with cylindrical symmetry.

3. What are the advantages of using cylindrical coordinates?

One advantage of using cylindrical coordinates is that they are well-suited for describing objects with cylindrical symmetry, such as cylinders, cones, and helices. They also make certain calculations, such as calculating the volume of a cylinder, simpler and more intuitive. Additionally, they can be easily converted to Cartesian coordinates if needed.

4. How are cylindrical coordinates used in real-world applications?

Cylindrical coordinates are used in a variety of real-world applications, such as in engineering, physics, and mathematics. They are particularly useful in fields such as fluid dynamics, electromagnetism, and solid mechanics. They are also commonly used in computer graphics and computer-aided design (CAD) software.

5. Are there any limitations to using cylindrical coordinates?

One limitation of using cylindrical coordinates is that they are not as versatile as Cartesian coordinates. They are most useful for describing objects with cylindrical symmetry and may not be as effective for representing more complex shapes. Additionally, some calculations may be more complicated in cylindrical coordinates compared to Cartesian coordinates.

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