How can we set up a triple integral to solve this problem?

In summary: The y-x plane is your domain:Here you can see how the y-x plane intersects the x-y plane at (0,0) and (1,1).
  • #1
stunner5000pt
1,461
2
Triple Integral setup...

[tex] \int \int \int_{G} 6x (z+y^3) dx dy dz [/tex] G bounded by [tex] x = 0, \ x = y, \ z = y-y^2, \mbox{and} \ z=y^2 - y^3 [/tex]
x from 0 to1
y from 0 to x
z from z=y-y^2 to y^2 - y^3
and the integration order becomes dz dy dx
would this give the right answer?

what aboiut this one
[tex] \int \int \int_{G} xy + xz dx dy dz [/tex]

G bounded by z = x, z=2-x, z = y^2
z goes from 2-y^2 to y^2
y goes from 2-x to x
x goes from 0 to 2
and the integration order to dz dy dx

I think the second one is wrong. Please do help!
 
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  • #2
Can you explain your reasoning for 1? Particularly the x and y. What exactly are you trying to do in these problems?
 
  • #3
the question is evaluate the integral over the given bounded region G

for the first one z goes like the y function, that s fine
for the y takes on a min value of 0 and max of x and
for the x the min value is zero but I am not sure about the max value ..
 
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  • #4
Think of x=0 and x=y as planes, not as lines.

Also, x takes on a min value of 0, and a max value of y. You don't know much about y.

Try imagining the region G in your head, its bounded by the yz plane, and the plane y=x and the z function.

You want to find how far y and x go given your constraints. Solving the functions would not be a bad idea.
 
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  • #5
if i were to solve what am i solving for?
which functions would i use?

x=0, x= y, z = y - y^2 , z = y^2 - y^3
the intesection of which surfaces??
 
  • #6
Theres only two equations that could be solved here. You have two functions 'z' in R^3. For a triple integral, you want to find the domain of these curves on the xy plane, where z=0. Solve the z functions to find the range of the y function.


http://tutorial.math.lamar.edu/AllBrowsers/2415/TripleIntegrals.asp Example 2
 
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  • #7
ok i got the y ranges from 0 to 1
now for the x part since x goes from 0 to y, x goes from 0 to 1
but y has to be a function of x so y goes from 0 to x?? and then x from 0 to 1?
 
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  • #8
[tex] \int\int \int_{G} 6x (z+y^3) dx dy dz [/tex] = [tex] \int\int_{D} \int_{y-y^2}^{y^2-y^3} 6x(z+y^3)dz dA[/tex]

http://www.public.asu.edu/~hyousif/maple.JPG
http://www.public.asu.edu/~hyousif/xzplane.JPG

edit: Its the yz plane, not the xz plane.

Heres the x-y plane, which is your domain:
Notice its a triangle with vertices at (0,0) (0,1) (1,1) You can express this as a domain pretty easily.

http://www.public.asu.edu/~hyousif/xyplane.JPG
 
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Related to How can we set up a triple integral to solve this problem?

What is the triple integral setup problem?

The triple integral setup problem involves calculating the volume of a three-dimensional region using a triple integral. This problem typically requires setting up the integral based on the given bounds and integrand.

What are the key steps in solving a triple integral setup problem?

The key steps in solving a triple integral setup problem include identifying the bounds of integration, determining the correct order of integration, setting up the integrand, and evaluating the integral using appropriate methods.

What are some common mistakes made when solving a triple integral setup problem?

Some common mistakes in solving a triple integral setup problem include using the incorrect order of integration, neglecting to include all necessary variables in the integrand, and incorrectly setting up the bounds of integration.

How can I improve my skills in solving triple integral setup problems?

To improve your skills in solving triple integral setup problems, it is important to practice regularly and familiarize yourself with different types of integrals and their corresponding bounds. You can also seek guidance from a teacher or textbook and use online resources to practice and check your solutions.

What real-world applications involve the use of triple integral setup problems?

Triple integral setup problems have various applications in physics, engineering, and economics, such as calculating the mass or volume of an object, determining the center of mass, finding the moment of inertia, and evaluating probability distributions.

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