Triple Integration from Rectangular to Spherical Coordinates

In summary: Actually the homework problem asked to convert to both cylindrical and spherical coordinates and I already finished the cylindrical. And yes thank you for cleaning up my equation!\int_{-2}^2\int_{-\sqrt{4-x^2}}^{\sqrt{4-x^2}}\int_{x^2+y^2}^4x\, dzdydxFirst, mark vertical lines, on an xy- graph, at x= -2 and x= 2. Of course, y= -\sqrt{4- x^2} and y= \sqrt{4- x^2} are halves of the circle
  • #1
enwarnock
2
0

Homework Statement



Convert the integral from rectangular coordinates to spherical coordinates

2 √(4-x^2) 4
∫ ∫ ∫ x dz dy dx
-2 -√(4-x^2) x^2+y^2

Homework Equations



x=ρ sin∅ cosθ
y=ρ sin∅ cosθ
z=ρ cos∅

In case the above integrals cannot be understood:
-2 ≤ x ≤ 2
-√(4-x^2) ≤ y ≤ √(4-x^2)
x^2+y^2 ≤ z ≤ 4

The Attempt at a Solution



I figured that 0≤θ≤2∏ but and that the x converts to ρ sin∅ cosθ
and I know you have to multiply the original converted function (ρ sin∅ cosθ) to ρ^2 sin∅
but that's all i figured out
 
Last edited:
Physics news on Phys.org
  • #2
enwarnock said:

Homework Statement



Convert the integral from rectangular coordinates to spherical coordinates

2 √(4-x^2) 4
∫ ∫ ∫ x dz dy dx
-2 -√(4-x^2) x^2+y^2

Homework Equations



x=ρ sin∅ cosθ
y=ρ sin∅ cosθ
z=ρ cos∅

The Attempt at a Solution



I figured that 0≤θ≤2∏ but and that the x converts to ρ sin∅ cosθ
and I know you have to multiply the original converted function (ρ sin∅ cosθ) to ρ^2 sin∅
but that's all i figured out

I assume that integtration means$$
\int_{-2}^2\int_{-\sqrt{4-x^2}}^{\sqrt{4-x^2}}\int_{x^2+y^2}^4x\, dzdydx$$The first thing you should do is draw a picture of the 3D region described by the limits. You will need it to do the spherical limits. Are you sure you aren't asked to put it into cylindrical coordinates? That would be the natural choice.
 
  • #3
LCKurtz said:
I assume that integtration means$$
\int_{-2}^2\int_{-\sqrt{4-x^2}}^{\sqrt{4-x^2}}\int_{x^2+y^2}^4x\, dzdydx$$The first thing you should do is draw a picture of the 3D region described by the limits. You will need it to do the spherical limits. Are you sure you aren't asked to put it into cylindrical coordinates? That would be the natural choice.

Actually the homework problem asked to convert to both cylindrical and spherical coordinates and I already finished the cylindrical.
And yes thank you for cleaning up my equation!
 
  • #4
[itex]\int_{-2}^2\int_{-\sqrt{4-x^2}}^{\sqrt{4-x^2}}\int_{x^2+y^2}^4x\, dzdydx[/itex]
First, mark vertical lines, on an xy- graph, at x= -2 and x= 2. Of course, [itex]y= -\sqrt{4- x^2}[/itex] and [itex]y= \sqrt{4- x^2}[/itex] are halves of the circle [itex]x^2+ y^2= 4[/itex] that lies between those vertical lines.

Then [itex]z= x^2+ y^2[/itex] to z= 4 can be written, in spherical coordinates, as [itex]\rho cos(\phi)= \rho^2 sin^2(\phi)[/itex] or [itex]\rho= cot(\phi)csc(\phi)[/itex] and [itex]\rho cos(\phi)= 4[/itex] or [itex]\rho= sec(\phi)[/itex]
 
  • #5
@enwarnock: So how are you coming on the spherical coordinate limits? Do you see that depending on what values ##\phi## takes that your ##\rho## is a two piece function and you are going to need two triple integrals to express the volume?
 

1. What is triple integration from rectangular to spherical coordinates?

Triple integration from rectangular to spherical coordinates is a mathematical process used to convert a triple integral, which is an integral with three variables, from rectangular coordinates to spherical coordinates. This is often done to make the integral easier to solve or to better understand the geometry of the problem.

2. Why would someone want to use spherical coordinates instead of rectangular coordinates?

Spherical coordinates are often used when dealing with problems that involve spherical or cylindrical symmetry, such as in physics or engineering. In these cases, spherical coordinates can simplify the problem and make it easier to solve. Additionally, spherical coordinates can sometimes provide a more intuitive understanding of the problem's geometry.

3. What are the steps for converting a triple integral from rectangular to spherical coordinates?

The first step is to rewrite the integral with the appropriate limits of integration in terms of spherical coordinates. Then, use the conversion formulas to express the integrand in terms of spherical coordinates. Next, evaluate the integral using the new limits and integrand. Lastly, convert the final answer back to rectangular coordinates if needed.

4. What are the conversion formulas for triple integration from rectangular to spherical coordinates?

The conversion formulas are:
x = ρsinφcosθ
y = ρsinφsinθ
z = ρcosφ
Where ρ is the distance from the origin to the point, φ is the angle between the positive z-axis and the line from the origin to the point, and θ is the angle between the positive x-axis and the projection of the point onto the xy-plane.

5. Are there any common mistakes to watch out for when performing triple integration from rectangular to spherical coordinates?

Yes, there are a few common mistakes to watch out for. One is forgetting to change the limits of integration when converting to spherical coordinates. Another is using the wrong conversion formulas, such as using the polar coordinates formulas instead of the spherical coordinates formulas. It is also important to be careful when evaluating the integral, as it can be easy to mix up the order of integration or make calculation errors.

Similar threads

  • Calculus and Beyond Homework Help
Replies
3
Views
474
  • Calculus and Beyond Homework Help
Replies
7
Views
649
  • Calculus and Beyond Homework Help
Replies
3
Views
2K
  • Calculus and Beyond Homework Help
Replies
9
Views
2K
  • Calculus and Beyond Homework Help
Replies
9
Views
1K
  • Calculus and Beyond Homework Help
Replies
3
Views
911
  • Calculus and Beyond Homework Help
Replies
4
Views
906
  • Calculus and Beyond Homework Help
Replies
4
Views
941
  • Calculus and Beyond Homework Help
Replies
1
Views
431
  • Calculus and Beyond Homework Help
Replies
5
Views
596
Back
Top