Triple Integration for Volume: Finding Intersections and Sketching Functions

AI Thread Summary
To find the volume between the two paraboloids defined by f(x,y) = x^2 + y^2 and g(x,y) = 20 - (x-4)^2 - (y+2)^2, it's essential to first determine their intersection. The intersection results in the equation (x+2)^2 + (y-1)^2 = 5, which describes a circle in the xy-plane. Once the intersection is established, the volume can be calculated by subtracting the two z-values and integrating over the defined circular region. For sketching the functions and finding intersections without graphing calculators, manual plotting of key points and symmetry considerations can be beneficial. Understanding these steps is crucial for accurately setting up the triple integral for volume calculation.
Tarhead
Messages
7
Reaction score
0
I have a group of problems that deals with the equations:

f(x,y)= x^2+y^2
g(x,y)=20-(x-4)^2-(y+2)^2

Can someone help find the triple integral to find the volume.
 
Physics news on Phys.org
I would start off by making at the very least a rough sketch of the volume you are trying to find, that way you can find out the boundaries you are dealing with.
 
It might help to clarify the problem: functions don't HAVE a volume!

If you mean "find the volume of the region bounded by z= x2+ y2 (a paraboloid) and z= 20- (x-4)2- (y+2)2 (also a paraboloid)" then you need to determine where the two paraboloids intersect and "project" that down to the xy-plane.

I get (x+2)2+ (y-1)2= 5, a circle. Subtract the two "z" values and integrate over that circle.

(Please do not post the same question twice!)
 
Last edited by a moderator:
Thanks for the help. I have problems with finding the intersection and projecting that on the x-y plane. We cannot use graphing calculators. Are there any easy ways to sketch the functions and/or find the intersection?
 

Thread 'Rolling without slipping on a curved surface'

Kindly see the attached pdf. My attempt to solve it, is in it. I'm wondering if my solution is right. My idea is this: At any point of time, the ball may be assumed to be at an incline which is at an angle of θ(kindly see both the pics in the pdf file). The value of θ will continuously change and so will the value of friction. I'm not able to figure out, why my solution is wrong, if it is wrong .
View full post »
Thread 'Voltmeter readings for this circuit with switches'

Thread 'Voltmeter readings for this circuit with switches'

TL;DR Summary: I would like to know the voltmeter readings on the two resistors separately in the picture in the following cases , When one of the keys is closed When both of them are opened (Knowing that the battery has negligible internal resistance) My thoughts for the first case , one of them must be 12 volt while the other is 0 The second case we'll I think both voltmeter readings should be 12 volt since they are both parallel to the battery and they involve the key within what the...
View full post »

Similar threads

The div in cartesian coordinates
Replies
2
Views
766
MHB Triple Integrals: Finding Volume of Solid S Bounded by Planes
Replies
5
Views
2K
Depth of a basketball floating on water
Replies
9
Views
917
Calculate the area of this pond with functions given for the perimeter
Replies
2
Views
803
Finding the shape of a hanging rope
Replies
3
Views
758
The volume of a "spherical cap" using triple integrals
Replies
9
Views
2K
Find the potential using a line integral (Electromagnetism)
Replies
1
Views
1K
Converting Area to Volume: A Spherical Coordinate Approach
Replies
2
Views
4K
Find the center of mass of a cone with variable density....
Replies
15
Views
6K
Find the linear speed of the particle when system rotates about axis
Replies
6
Views
1K

Hot Threads

Recent Insights

Back
Top