Understanding Integration Limits for Spherical and Cartesian Coordinates

In summary, the conversation discusses the integration of a sphere in both spherical and cartesian coordinates. The question also mentions translating the origin to the shape centre and checking the limits of integration. The attempted solution includes finding the limits of integration and a factor of 2 that is unclear.
  • #1
MoAli
12
0

Homework Statement


Shown in the photo attached.
Screen Shot 2017-02-12 at 11.05.57.png


2. Homework Equations
∫V r2Sinθdθdφdr in spherical coordinates
∫V dxdydz in cartesian coordinates
equation of a sphere x2+y2+z2=r2

The Attempt at a Solution


In this case y=(y-2): sphere displaced on the y-axis. and since it is bound by all planes its going to be one quarter of a sphere. I don't get the part where the question says translate the origin to the shape centre, how can I do this? and also I need someone to check my limits of integration. I attached my answer.
IMG_7777.JPG
 
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  • #2
Hi,

Can't say I understand your integration limits for the second part. Care to explain ? And where does the factor 2 come from ?
 

FAQ: Understanding Integration Limits for Spherical and Cartesian Coordinates

What is triple integration and when is it used?

Triple integration is a mathematical technique used to find the volume of a three-dimensional region. It is also used to calculate the mass, center of mass, and moment of inertia of an object. It is commonly used in physics, engineering, and other fields that involve three-dimensional calculations.

How do you set up a triple integration problem?

To set up a triple integration problem, you first need to identify the bounds for each variable. This involves finding the limits of integration for each variable based on the given region. Then, you can choose the order of integration, which is typically from inside to outside (e.g. dx dy dz). Finally, you can write the appropriate integrals using the bounds and the function to be integrated.

What are some common applications of triple integration?

As mentioned, triple integration is commonly used in physics and engineering for calculating volume, mass, and moments of inertia. It is also used in economics for finding the area under a three-dimensional demand or supply curve. In computer graphics, it is used for calculating lighting and shading effects. Triple integration is also used in geology for calculating the volume of geological structures.

What is the difference between triple integration and single or double integration?

The main difference between triple integration and single or double integration is the number of dimensions. Single integration deals with one-dimensional problems (e.g. finding the area under a curve), while double integration deals with two-dimensional problems (e.g. finding the volume under a surface). Triple integration, on the other hand, deals with three-dimensional problems and is more complex because it involves integrating over three variables instead of one or two.

Are there any tips or tricks for solving triple integration problems?

One tip for solving triple integration problems is to draw a diagram of the region to help visualize the bounds and the order of integration. Another tip is to choose the order of integration based on the complexity of the bounds and the function to be integrated. In some cases, changing the order of integration may make the problem easier to solve. It is also important to carefully evaluate the integrals and check for any mistakes in the limits of integration or the integrand.

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