Help please in understanding the limits of this integration

In summary: So if you want to integrate over that region, you need to integrate from a lower limit that the ##r##-value at the intercept, not from a higher limit.In summary, the problem involves finding the volume above the cone z = sqrt(x^2 = y^2) and below the sphere x^2 + y^2 + z^2 = 1. The solution manual uses radial limits from 0 to sqrt(1/2) instead of sqrt(1/2) to 1, which may seem counterintuitive. However, this is because the region above the cone and below the sphere has a lower r-value at the intercept, so the integral must be evaluated from a lower limit. This can also be
  • #1
snatchingthepi
148
38
Homework Statement
Finding the volume above the cone z = sqrt(x^2 = y^2) and below sphere x^2 + y^2 + z^2 = 1
Relevant Equations
Cone: z = sqrt(x^2 = y^2)

Sphere: x^2 + y^2 + z^2 = 1
So I can push this integral all the way to the end and see I get a negative volume.

I solve for the intercepts of the cone and sphere at r^2 = 1/2. Seeing this cone is inside the sphere and the sphere is around it, I figure I should integrate from sqrt(1/2) to 1 since we're dealing with a unit sphere, and these limits would give the area *above* the cone and *below* the sphere.

But when I check the solution manual I see the problem involves using radial limits from 0 to sqrt(1/2) but the equation that is integrated is exactly the same as mine. Using limits from 0 to root(1/2) would seem to me to give the area above the plane projection of the sphere and below the cone, not above the cone and below the sphere. Can someone help me understand where this is coming from?
 
Physics news on Phys.org
  • #2
snatchingthepi said:
Problem Statement: Finding the volume above the cone z = sqrt(x^2 = y^2) and below sphere x^2 + y^2 + z^2 = 1
Relevant Equations: Cone: z = sqrt(x^2 = y^2)

Sphere: x^2 + y^2 + z^2 = 1

So I can push this integral all the way to the end and see I get a negative volume.

I solve for the intercepts of the cone and sphere at r^2 = 1/2. Seeing this cone is inside the sphere and the sphere is around it, I figure I should integrate from sqrt(1/2) to 1 since we're dealing with a unit sphere, and these limits would give the area *above* the cone and *below* the sphere.

But when I check the solution manual I see the problem involves using radial limits from 0 to sqrt(1/2) but the equation that is integrated is exactly the same as mine. Using limits from 0 to root(1/2) would seem to me to give the area above the plane projection of the sphere and below the cone, not above the cone and below the sphere. Can someone help me understand where this is coming from?
Can you show us the integral you're evaluating? Your description omits some details that would be helpful.
 
  • #3
Does this plot help?
Untitled.png
 
  • #4
245610


Here is my image of the integral I've computed. Barring an extra parentheses I've clearly dropped by accident at the end, I am trying to run it from sqrt(1/2) to 1. Looking at the graph above provided by vela (thank you) I see where everything intercepts, but I do not understand why the given solutions integrates from 0 to sqrt(1/2).

To get the area above the cone and below the sphere shouldn't I have to use limits of integration from sqrt(1/2) to 1?
 
  • #5
First of all, this is (much) easier to do in spherical coordinates than in cylinder coordinates.

Second, in the region that is above the yellow line and below the blue line in #3, clearly the ##r##-value is lower than the ##r##-value at the intercept.
 

1. What is integration in science?

Integration in science refers to the process of combining different fields of study or disciplines to gain a deeper understanding of a particular topic or problem. It involves synthesizing information and concepts from various sources to create a more comprehensive understanding.

2. Why is understanding the limits of integration important?

Understanding the limits of integration is important because it allows scientists to recognize the boundaries and constraints of their research. This helps them to accurately interpret their findings and avoid making false conclusions.

3. What are the common challenges in integrating different fields of study?

Some common challenges in integration include differences in terminology, methods, and theories between different fields. There may also be conflicting findings or interpretations that need to be reconciled for a cohesive understanding.

4. How do scientists determine the appropriate scope of integration for their research?

Determining the appropriate scope of integration depends on the specific research question and the available evidence. Scientists must carefully consider the relevance and reliability of information from different fields before integrating it into their study.

5. Can integration lead to biased or incomplete conclusions?

Yes, integration can potentially lead to biased or incomplete conclusions if not done carefully and objectively. It is important for scientists to critically evaluate and validate the information from different fields before integrating it into their research.

Similar threads

  • Calculus and Beyond Homework Help
Replies
3
Views
942
  • Calculus and Beyond Homework Help
Replies
3
Views
547
  • Calculus and Beyond Homework Help
Replies
24
Views
2K
  • Calculus and Beyond Homework Help
Replies
5
Views
1K
  • Calculus and Beyond Homework Help
Replies
21
Views
2K
  • Calculus and Beyond Homework Help
Replies
12
Views
1K
  • Calculus and Beyond Homework Help
Replies
16
Views
1K
  • Calculus and Beyond Homework Help
Replies
5
Views
787
  • Calculus and Beyond Homework Help
Replies
14
Views
200
  • Calculus and Beyond Homework Help
Replies
13
Views
1K
Back
Top