Finding the bounds of a triple integral in cylindrical coordinates?

In summary, the student is trying to solve a problem in cylindrical coordinates, but is confused about the y bounds. He recommends drawing the limits and noting that the range of Theta is from 0 to pi/4.
  • #1
Lauren72
5
0

Homework Statement



I took a picture of the problem so it would be easier to understand.

Cylindrical.png


All I need to know is what the bounds are.

Homework Equations



In cylindrical:

x=rcos(theta)
y=rsin(theta)
z=z

The Attempt at a Solution



I don't know why we should change this to cylindrical. I feel like it's easier in cartesian. I solved it the normal way and easily got -160/3. But, if we must do cylindrical I already know what the function is. It's just r^2cos^2(Theta)z. What I cannot figure out is the bounds. I'm really confused because of the y bounds. When I try to draw the image I get the parabola formed by the first bound, but I can't figure out how it intersects the rest of the limits, particularly y=x. How do you find those bounds? What are they?

Please help! I'm studying for my Calc III final on Monday.
 
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  • #2
It may well be easier to evaluate in Cartesian coordinates, but that is not the point of the problem! The point is precisely to practice changing from one coordinate system to another.

I recommend drawing the given limits. The "outer" integral, with respect to x, has lower limit 0 and upper limit 2. Okay, draw xy-axes, draw vertical lines at x= 0 (the y-axis) and at x= 2. That marks the "left" and "right" bounds of the area. The "middle" integral, with respect to y, has limits of 0 and x. Draw the horizontal line y= 0 (the x-axis) and the line y= x. You should see that that forms a triangle with vertices at (0, 0), (2, 0), and (2, 2). The "outer" integral, with respect to z, has limits 0 and [itex]4- x^2[/itex] so the lower boundary is the xy-plane and the upper boundary is the "parabolic cylinder" given by [itex]z= 4- x^2[/itex] (since there is no "y" in the formula, the parabola extends up the y- direction).

However, since you only want to change to cylindrical coordinates, you don't need to worry changing that, except to note that [itex]x= r cos(\theta)[/itex] so [itex]4- x^2= 4- r^2 cos^2(\theta)[/itex]. The "inner" integral, with respect to z, will be from 0 to [itex]4- r^2 cos^2(\theta)[/itex].

Now look at that triangle. Isn't it easy to see that the angle, [itex]\theta[/itex] ranges from x-axis to the line y= x? What angles are those? And, for each angle, r ranges from the origin (r= 0, of course) to the line x= 2. Since [itex]x= r cos(\theta)[/itex], that is the same as [itex]r= 2/cos(\theta)= 2 sec(\theta)[/itex].
 
  • #3
Okay. So Theta ranges from 0 to y=x, therefore it goes from 0 to pi/4, yes?

So when you're finding the bounds for r and theta, is it necessary to draw the 3d graph? It looks like you can find the bounds just by drawing it on the xy plane.

And since we're in cyclindrical, and not spherical, that's why we don't have to worry about doing anything strange to the z bounds, right? We just do a direct substitution?
 

1. What are cylindrical coordinates?

Cylindrical coordinates are a type of coordinate system used to describe the position of a point in three-dimensional space. They consist of a radius, an angle, and a height, and are often represented as (r,θ,z). They are commonly used in mathematics and physics, particularly in situations involving cylindrical shapes or symmetry.

2. How do I convert between Cartesian and cylindrical coordinates?

To convert from Cartesian coordinates (x,y,z) to cylindrical coordinates (r,θ,z), you can use the following formulas:

r = √(x^2 + y^2)

θ = arctan(y/x)

z = z

To convert from cylindrical coordinates to Cartesian coordinates, you can use these formulas:

x = rcos(θ)

y = rsin(θ)

z = z

3. What is the volume element in cylindrical coordinates?

The volume element in cylindrical coordinates is r dr dθ dz. This is because the volume of a small element in cylindrical coordinates is equal to the product of its height (dz), its radius (r), and its circumference (2πr dθ).

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

To set up a triple integral in cylindrical coordinates, you will need to determine the bounds for each variable (r, θ, and z). This will depend on the shape and limits of the region you are integrating over. Once you have determined the bounds, you can write the integral as:

∫∫∫f(r,θ,z) r dr dθ dz

5. How do I find the bounds of a triple integral in cylindrical coordinates?

To find the bounds of a triple integral in cylindrical coordinates, you can use the following steps:

  1. Sketch the region you are integrating over in cylindrical coordinates.
  2. Determine the limits for each variable (r, θ, and z) by looking at the boundaries of the region.
  3. Set up the integral using the bounds you have determined.
  4. Solve the integral to find the desired value.

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