Finding Volume Inside Sphere and Cone: Limits of Integration Explained

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In summary, the problem is to find the volume inside a sphere and a cone, given their equations. The solution involves triple integration in spherical coordinates and the final answer is (2/3)*pi*1^3(1-1/sqrt(2)). The limits of integration for theta are (0,2pi) and for phi are (0, pi/4). The reasoning for phi's limits is explained as it represents the co-latitude angle from the z-axis to the line z=x.
  • #1
stratusfactio
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Homework Statement


The textbook I have is horrible in explaining this stuff in the chapter so please bare with me.

So we have the information: Find the volume V inside both the sphere x^2 + y^2 + z^2 = 1 and the cone z =sqrt(x^2 + y^2)


Homework Equations


Where the heck do I began? I know this will require triple integration in spherical coordinates and the solution is (2/3)*pi*1^3(1-1/sqrt(2))



The Attempt at a Solution


I know the theta integrand is (0,2pi) because there are no bounds, but I have no idea where to find the limits of integration for the phi and zenith integrands. :(

EDIT (1): Just solved for the intersections of the sphere and cone and got z = 1/sqrt(2))
EDIT (2): Since phi = 1 and z = (1/sqrt(2)), it's safe to conclude that the zenith angle is pi/4...but what would be it's upper limit?
EDIT (3): I believe the limits for phi is 0 to pi/4. But I'm not quite sure why though? I think I got the integrands all figured out. I believe it's: for p = (0,1), phi = (0, pi/4) and theta = (0,2pi).

So overall, could someone just briefly explain to me why phi's limits of integration is from 0 to pi/4. thanks a bunch :D
 
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  • #2
Looking at the figure from the side (along the negative y-axis), you should see a circle and a "V" (y= 0 so the equations are [itex]x^2+ z^2= 1[/itex], a circle, and [itex]z= \sqrt{x^2}= |x|[/itex]). The "co-latitude", [itex]\phi[/itex], goes from directly up the z-axis, [itex]\phi= 0[/itex], to the line z= x, which bisects the right angle between the x and z axes and so makes angle [itex]\pi/4[/itex] with the z-axis.
 
  • #3
stratusfactio said:

Homework Statement


The textbook I have is horrible in explaining this stuff in the chapter so please bare with me.

I'm sorry. Undressing is forbidden in this forum. :eek:
 

What is volume and how is it measured?

Volume is the amount of space occupied by a three-dimensional object. It is measured in units such as cubic meters (m3) or cubic centimeters (cm3) depending on the size of the object.

What is the formula for finding volume?

The formula for finding volume depends on the shape of the object. For example, the formula for a cube is V = s3, where s is the length of one side. The formula for a cylinder is V = πr2h, where r is the radius of the base and h is the height.

How do you find the volume of irregularly shaped objects?

To find the volume of an irregularly shaped object, you can use the water displacement method. Fill a graduated cylinder with a known amount of water, then submerge the object in the water. The difference in the water level before and after submerging the object will give you the volume of the object.

Why is finding volume important in science?

Finding volume is important in science because it helps us understand the physical properties of objects and how they relate to each other. It is also a crucial measurement in fields such as chemistry, physics, and engineering.

What are some real-world applications of finding volume?

Finding volume is used in many real-world applications, such as calculating the amount of liquid in a container, determining the capacity of a building or room, and measuring the amount of gas in a tank. It is also used in fields like architecture and product design to create accurate models and prototypes.

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