Triple Integral Limits Help. Cylindrical Coordinates

In summary, for the first problem, we need to find the volume of the solid bounded by two paraboloids, z=x^2+y^2 and z=36-x^2-y^2, which can be expressed as 324π.For the second problem, we need to use cylindrical coordinates to find the volume inside the region cut from the sphere x^2+y^2+z^2=36 by the cylinder x^2+y^2=4. The correct answer is \frac{4\pi}{3}(216-32^{3/2}), and the setup for the integral should be V=\int_{0}^{2\pi} \int_0^2 \int_{-\sqrt{36
  • #1
bob29
18
0

Homework Statement


Find the volume of the solid bounded by the paraboloids z=x^2+y^2 and z=36-x^2-y^2.

Answer is:
[tex]
324\pi
\\[/tex]

Homework Equations


r^2=x^2+y^2
x=rcos0
y=rcos0

The Attempt at a Solution


36-x^2+y^2=x^2+y^2\\
36=2x^2+2y^2
18=x^2+y^2
r^2=18

[tex]
V=\int_{0}^{2\pi} \int_0^{3\sqrt{2}} \int_{r^2}^{36-r^2} \left (1) \right dz.rdr.d\theta
[/tex]

---------------------------------------------------------------

Homework Statement


Use cylindrical coordinates to find the volume of inside the region cut from the sphere x^2+y^2+z^2=36 by the cylinder x^2+y^2=4.
Ans is:
[tex]
\frac{4\pi}{3}(216-32^{3/2})
[/tex]

The Attempt at a Solution


Diagram is of a sphere where a cylinder is inside of it.
x^2+y^2+z^2=4
r^2+z^2=4
[tex]
z=\sqrt{4-r^2}\\
[/tex]

x^2+y^2=4
r^2=2^2
r=2

[tex]
V=\int_{0}^{2\pi} \int_0^2 \int_{-\sqrt{36-r^2}}^{\sqrt{4-r^2}} \left (1) \right dz.rdr.d\theta
[/tex]

Studying for an exam and would appreciate the help to answering these questions that I am struggling on.
 
Last edited:
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  • #2
The first problem looks good: like you I got [itex] 324 \pi [/itex].

Your setup for the second volume is almost correct, but not quite. Your upper limit of integration for z is incorrect. When you're picturing the region, remember that it's the sphere that gives you the upper and lower boundaries for the z coordinate.
 
  • #3
Isn't the region the cylinder? so it enters the bottom and exits the top of the cylinder. Sort of confused about which function determines the boundaries.

Thanks for confirming the 1st problem.
Guess I made a mistake in my integration of r where r^4/4 should have been r^4/2.
 

1. What are cylindrical coordinates?

Cylindrical coordinates are a type of coordinate system used to describe points in three-dimensional space. They consist of a radial distance from the origin, an angle in the xy-plane, and a height along the z-axis.

2. How do I convert from rectangular to cylindrical coordinates?

To convert from rectangular coordinates (x, y, z) to cylindrical coordinates (r, θ, z), use the following equations:

r = √(x² + y²)

θ = tan⁻¹(y/x)

z = z

3. What are the limits of integration for a triple integral in cylindrical coordinates?

The limits of integration for a triple integral in cylindrical coordinates depend on the shape of the region being integrated over. Generally, the limits for r will be a function of θ, and the limits for θ and z will be constants.

4. How do I set up a triple integral in cylindrical coordinates?

To set up a triple integral in cylindrical coordinates, first determine the limits of integration for each variable. Then, use the appropriate integrand for the shape of the region being integrated over. The integral should be written in the form ∫∫∫f(r, θ, z) rdrdθdz.

5. Can you provide an example of evaluating a triple integral in cylindrical coordinates?

Sure! Let's say we want to find the volume of a cylinder with radius 2 and height 5. The limits of integration would be r = 0 to 2, θ = 0 to 2π, and z = 0 to 5. The integrand would be 1 (since we are just looking for the volume). This would result in the integral ∫∫∫ 1 rdrdθdz, which evaluates to 20π.

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