Find the triple integral of xy

In summary, the conversation is about finding the triple integral of xy where E is bounded by the curves y = x^2 and x = y^2, as well as the planes z = 0 and z = x + y. The correct approach is to use the region [0,1] x [0,1] x [0,x+y] and evaluate the triple integral dzdydx. The correct limits for the second integral are the upper and lower limits of y for a constant value of x. In this case, the lower bound should be x^2 and the upper bound should be sqrt(x), as the region is only in the first quadrant.
  • #1
magnifik
360
0
find the triple integral of xy where E is bounded by y = x^2 and x = y^2 and the planes z = 0 and z = x + y.

i got 1/3 as a solution, but I'm not sure if i did it right, specifically the part in finding the boundaries for x and y. i found that they intersected at (0,0) and (1,1) so i had the region
[0,1] x [0,1] x [0,x+y] then evaluated the triple integral dzdydx. did i take the right approach?
 
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  • #2
Your limits for the second integral are incorrect. If your outside integral is with respect to x, the limits of the middle integral should be the upper and lower limits of y for a constant value of x.
 
  • #3
vela said:
Your limits for the second integral are incorrect. If your outside integral is with respect to x, the limits of the middle integral should be the upper and lower limits of y for a constant value of x.

is the boundary 0 to x^2?
 
  • #4
No. Graph the functions on the xy-plane.
 
  • #5
since it's only in the first quadrant the lower bound should be x^2 and upper bound should be sqrt(x)?
 
  • #6
Yup, that's right.
 

1. What is the purpose of finding the triple integral of xy?

The triple integral of xy is used to calculate the volume under a surface in three-dimensional space. It is a key concept in multivariable calculus and has applications in physics, engineering, and other fields.

2. How is the triple integral of xy calculated?

The triple integral of xy is calculated by integrating the function xy over a three-dimensional region, typically denoted as ∭xy dV. This involves breaking down the region into smaller, simpler shapes and using the appropriate integration techniques for each shape.

3. What are the limits of integration for finding the triple integral of xy?

The limits of integration for finding the triple integral of xy depend on the shape of the region being integrated over. For rectangular regions, the limits are the minimum and maximum values of x, y, and z. For more complex regions, the limits may need to be determined by using equations or geometric principles.

4. What are some applications of the triple integral of xy?

The triple integral of xy has many applications in physics and engineering, such as calculating the center of mass of a three-dimensional object, determining the mass of a solid with varying density, and finding the electric field produced by a three-dimensional charge distribution.

5. How does finding the triple integral of xy relate to other concepts in calculus?

The triple integral of xy is closely related to other concepts in calculus, such as double integrals and single integrals. It is also connected to the fundamental theorem of calculus, which states that integration is the reverse process of differentiation. Furthermore, the triple integral is a fundamental tool in vector calculus, which is used to study and model physical systems in three-dimensional space.

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