Integrating to find the volume of a finite region

In summary, the conversation discusses finding the volume of a finite region enclosed by the surfaces z = 0 and x2 + y2 + z = 1. It is suggested to use triple integration, but the conversation suggests a simpler method involving a single integration. The first step is to sketch the region and then slice it into thin slices to determine its area.
  • #1
james525
1
0
Find the volume of the finite region enclosed by the surfaces z = 0 and
x2 + y2 + z = 1

I know I have to do triple integration on dV to accomplish this but do not know where to start and what limits to use for x, y and z?

Cheers guys
 
Last edited:
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  • #2
welcome to pf!

hi james! welcome to pf! :smile:

(try using the X2 icon just above the Reply box :wink:)
james525 said:
I know I have to do triple integration on dV to accomplish this but do not know where to start and what limits to use for x, y and z?

no, it's symmetric, so you can get away with a single integration! :wink:

first sketch the region (so that you know what it looks like), then slice it into very thin slices whose area you already know :smile:
 

1. What is integration?

Integration is a mathematical concept that involves finding the area under a curve. It is essentially the reverse process of differentiation.

2. How is integration used to find the volume of a finite region?

To find the volume of a finite region, we use a technique called "volume by cross-section". This involves dividing the region into infinitesimally thin slices, finding the area of each slice using integration, and then summing up all the areas to get the total volume.

3. Can integration be used for irregularly shaped regions?

Yes, integration can be used for irregularly shaped regions as long as we can express the boundaries of the region as a function or functions. We can then apply the same technique of volume by cross-section to find the volume of the region.

4. What are the limits of integration and how do we determine them?

Limits of integration refer to the boundaries of the region in terms of the independent variable. These can be determined by looking at the given region and identifying the points where the region starts and ends in terms of the independent variable. These points will be the lower and upper limits of integration, respectively.

5. Are there any limitations to using integration to find the volume of a finite region?

Yes, integration can only be used to find the volume of regions that can be expressed as a function or functions. It also requires some knowledge of calculus and can be a time-consuming process for complex regions. Additionally, it may not be applicable for regions with changing density.

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