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Analiza matematyczna funkcji wielu zmiennych

Kurs: Analiza matematyczna funkcji wielu zmiennych > Rozdział 4

Lekcja 8: Całkowanie we współrzędnych krzywoliniowych: współrzędne biegunowe, współrzędne cylindryczne i współrzędne sferyczne.
Matematyka
Analiza matematyczna funkcji wielu zmiennych
Całkowanie funkcji wielu zmiennych
Całkowanie we współrzędnych krzywoliniowych: współrzędne biegunowe, współrzędne cylindryczne i współrzędne sferyczne.
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Całkowanie we współrzędnych walcowych

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Wyrażanie całki po danym obszarze lub w zadanych granicach we współrzędnych walcowych. Tłumaczenie na język polski: Fundacja Edukacja dla Przyszłości.

Kontekst

Do czego zmierzamy

  • The main thing to remember about triple integrals in cylindrical coordinates is that start color #bc2612, d, V, end color #bc2612, representing a tiny bit of volume, is expanded as
    start color #bc2612, d, V, end color #bc2612, equals, r, d, theta, d, r, d, z
    (Don't forget to include the r)
  • Using cylindrical coordinates can greatly simplify a triple integral when the region you are integrating over has some kind of rotational symmetry about the z-axis.

The one rule

When performing double integrals in polar coordinates, the one key thing to remember is how to expand the tiny unit of area start color #bc2612, d, A, end color #bc2612 in terms of d, r and d, theta
∬Rf(r,θ) dA=∬Rf(r,θ) r dθ dr\begin{aligned} \iint_R f(r, \theta)\,\redE{dA} = \iint_R f(r, \theta)\,\redE{r\,d\theta\,dr} \end{aligned}
Note that the variable r is part of this expansion. Expanding the tiny unit of volume d, V in a triple integral over cylindrical coordinates is basically the same, except that now we have a d, z term:
∭Rf(r,θ,z) dV=∭Rf(r,θ,z) r dθ dr dz\begin{aligned} \iiint_R f(r, \theta, z)\,\redE{dV} = \iiint_R f(r, \theta, z)\,\redE{r\,d\theta\,dr\,dz} \end{aligned}
Remember, the reason this little r shows up for polar coordinates is that a tiny "rectangle" cut by radial and circular lines has side lengths r, d, theta and d, r.
The key thing to remember here is that theta is not a unit of length, so d, theta does not represent a tiny length in the same way that d, r and d, z do. It measures radians, which need to be multiplied by the distance r from the origin to become a length.

Example 1: Volume of a sphere

Problem: Find the volume of a sphere with radius 1 using a triple integral in cylindrical coordinates.
First of all, to make our lives easy, let's place the center of the sphere on the origin.
Next, I'll give the sphere a name, S, and write the abstract triple integral to find its volume.
∭SdV=∭Sr dθ dr dz\begin{aligned} \iiint_S \redE{dV} = \iiint_S \redE{r\,d\theta\,dr\,dz} \end{aligned}
As always, the hard part is putting bounds on the integral. However, doing this with cylindrical coordinates is much easier than it would be for cartesian coordinates. In particular, r and theta will just live within the unit disc, which is very natural to describe in polar coordinates:
Concept check: Which of the following sets of bounds for r and theta should we use to integrate over the unit disc?
Wybierz 1 odpowiedź:

Since the bounds of z will depend on the value of r, we let the innermost integral handle z, while the outer two integrals take care of r and theta. Writing down what we have so far, we get
∫02π∫01∫??r dz dr dθ\begin{aligned} \int_0^{2\pi} \int_0^1 \int_{?}^{?} r \,dz \,dr \,d\theta \end{aligned}
Remember, it's important to make sure the order of the differential terms d, z, d, r and d, theta matches up with the appropriate integral.
This next question is a little trickier.
Concept check: For a given value of r, which of the following shows the right range of values for z?
Wybierz 1 odpowiedź:

Applying this bound to our innermost integral, we get something that can be worked out.
Concept check: Solve this triple integral.
∫02π∫01∫−1−r21−r2r dz dr dθ=\begin{aligned} \int_0^{2\pi} \int_0^1 \int_{-\sqrt{1-r^2}}^{\sqrt{1-r^2}} r \,dz \,dr \,d\theta = \end{aligned}

And with that, you just found the volume of a unit sphere!
Moreover, this tool is powerful enough to do more than just find the volume of the sphere. For example, you could integrate a three-variable function f, left parenthesis, r, comma, theta, comma, z, right parenthesis inside the sphere,
∫02π∫01∫−1−r21−r2f(r,θ,z)r dz dr dθ\begin{aligned} \int_0^{2\pi} \int_0^1 \int_{-\sqrt{1-r^2}}^{\sqrt{1-r^2}} f(r, \theta, z) r \,dz \,dr \,d\theta \end{aligned}
The hard part of finding the bounds is no different, but the computation of the integrals (done by either you or a computer) will change.

Example 2: Integrating over a pie slice

For this example, we will integrate over a region which looks kind of like a slanted pie slice:
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In a problem, this region might be described to you using the following list of properties:
  • x, is greater than or equal to, 0
  • y, is greater than or equal to, 0
  • z, is greater than or equal to, 0
  • y, is less than or equal to, x
  • x, squared, plus, y, squared, is less than or equal to, 4
  • z, is less than or equal to, start fraction, y, divided by, x, end fraction
This time, we will not just be finding the volume of this region. Instead, our task is to integrate the following three-variable function:
f, left parenthesis, x, comma, y, comma, z, right parenthesis, equals, z, minus, x, squared, minus, y, squared
This might seem out of place in an article about integrating in cylindrical coordinates, since everything here is given in cartesian coordinates. Indeed, you could setup the triple integral using cartesian coordinates if you wanted. However, there's one key fact suggesting that our lives can be made dramatically easier by converting to cylindrical coordinates first:
  • The expression x, squared, plus, y, squared shows up in the function f, as well as in the description of the bounds. This suggests some rotational symmetry around the z-axis, which cylindrical coordinates are well-suited for.
For example, look at the range for our x and y values:
  • x, is greater than or equal to, 0
  • y, is greater than or equal to, 0
  • y, is less than or equal to, x
  • x, squared, plus, y, squared, is less than or equal to, 4
Describing this with a pair of integrals over d, x and d, y is a real pain. However, in polar coordinates, this becomes very simple:
  • 0, is less than or equal to, theta, is less than or equal to, start fraction, pi, divided by, 4, end fraction
  • 0, is less than or equal to, r, is less than or equal to, 2
This means the bounds on the integrals handling d, theta and d, r will be constants. You can't do better than that!
What about the other criteria, such as
  • z, is less than or equal to, start fraction, y, divided by, x, end fraction
Since converting to polar coordinates involves the property
tangent, left parenthesis, theta, right parenthesis, equals, start fraction, y, divided by, x, end fraction
The bounds on z can be translated to
  • 0, is less than or equal to, z, is less than or equal to, tangent, left parenthesis, theta, right parenthesis
Putting this together, our triple integral looks like this:
∫0π/4∫02∫0tan⁡(θ)f dV\begin{aligned} \int_0^{\pi/4} \int_0^2 \int_0^{\tan(\theta)} f\,\redE{dV} \end{aligned}
Notice how simple the bounds are. If you are up for a little pain, you can try finding the appropriate triple integral bounds in cartesian coordinates to see just how much uglier they are.
We now write the function f using polar coordinates.
f(x,y,z)=z−x2−y2⇓f(r,θ,z)=z−r2\begin{aligned} f(x, y, z) &= z - x^2 - y^2 \\ &\Downarrow \\ f(r, \theta, z) &= z - r^2 \end{aligned}
And of course, we incorporate the main takeaway of this article, which is how to write start color #bc2612, d, V, end color #bc2612 in polar coordinates:
start color #bc2612, d, V, end color #bc2612, equals, r, d, theta, d, r, d, z
Putting this all together, we get our triple integral in its final solvable state.
More practice: Solve this integral
∫0π/4∫02∫0tan⁡(θ)(z−r2)r dz dr dθ=\begin{aligned} \int_0^{\pi/4} \int_0^2 \int_0^{\tan(\theta)} (z - r^2)r \,dz \,dr \,d\theta = \end{aligned}

Podsumowanie

  • The main thing to remember about triple integrals in cylindrical coordinates is that start color #bc2612, d, V, end color #bc2612, representing a tiny bit of volume, is expanded as
    start color #bc2612, d, V, end color #bc2612, equals, start color #bc2612, r, d, theta, d, r, d, z, end color #bc2612
    (Don't forget to include the r)
  • Using cylindrical coordinates can greatly simplify a triple integral when the region you are integrating over has some kind of rotational symmetry about the z-axis.

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