Is the Triple Integral in Cylindrical Coordinates Correctly Solved?

In summary, the conversation discusses the use of cylindrical coordinates to solve a problem involving a region E and the evaluation of its volume. There is some confusion about the boundaries of integration and whether the method used is correct. The summary also mentions a correction made by one of the speakers and confirmation that the solution is correct. The conversation ends with a statement about 1008 being divisible by 3.
  • #1
daphnelee-mh
66
4
Homework Statement
(question is attached below)
Relevant Equations
x^2+y^2=r^2
dV=rdrdzdΘ
1593762965635.png


1593762987337.png


I am trying to solve it using cylindrical coordinates, but I am not sure whether the my description of region E is correct, whether is the value of r is 2 to 4, or have to evaluate the volume 2 times ( r from 0 to 4 minus r from 0 to 2), and whether is okay to take z from r^2/2 to 8
 
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  • #2
[tex]\int_2^8 dz \int_0^{2z} 2\pi \rho^3 d\rho[/tex]
 
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  • #3
May I know which method did you use? I have no idea of it, thanks
 
  • #4
Find attached my handwriting.
 

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  • #5
anuttarasammyak said:
[tex]\int_2^8 dz \int_0^{2z} 2\pi \rho^3 d\rho[/tex]
Shouldn't the boundary of integration be ##\sqrt{2z}## instead of ##2z##?
 
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  • #6
Yes. Thanks for correction.
 
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  • #7
1593775272308.png

Can you help to see whether this is correct ? Thank you
 
  • #8
Yes it looks correct to me.
 
  • #9
Thank you so much
 
  • #10
daphnelee-mh said:
View attachment 265726
Can you help to see whether this is correct ? Thank you
The digits of 1008 add to 9, so...
 
  • #11
haruspex said:
The digits of 1008 add to 9, so...
?
 
  • #12
daphnelee-mh said:
?
So 1008 is divisible by 3.
 
  • #13
haruspex said:
So 1008 is divisible by 3.
okay , thank you
 

What is a triple integral?

A triple integral is a mathematical concept used to calculate the volume of a three-dimensional shape. It involves integrating a function over a three-dimensional region.

How do you solve a triple integral?

To solve a triple integral, you need to first determine the limits of integration for each variable. Then, you can use the appropriate integration techniques, such as the method of cylindrical shells or the method of cross-sections, to evaluate the integral.

What are the applications of triple integrals?

Triple integrals are commonly used in physics, engineering, and other fields to calculate quantities such as mass, center of mass, and moments of inertia for three-dimensional objects.

What is the difference between a definite and indefinite triple integral?

A definite triple integral has specific limits of integration, while an indefinite triple integral does not. A definite triple integral gives a numerical value, while an indefinite triple integral gives a function of the remaining variables.

What are some tips for solving a difficult triple integral?

Some tips for solving a difficult triple integral include breaking it down into smaller parts, using symmetry to simplify the integral, and using appropriate substitution or change of variables to make the integral easier to evaluate.

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