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Topics You Need To Understand For This Page
Trig Identities and Formulae - NEW
basic trig identities |
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\(\sin^2\theta+\cos^2\theta=1\) | \(1+\tan^2\theta=\sec^2\theta\) |
\(\displaystyle{\tan\theta=\frac{\sin\theta}{\cos\theta}}\) | \(\displaystyle{\cot\theta=\frac{\cos\theta}{\sin\theta}}\) |
\(\displaystyle{\sec\theta=\frac{1}{\cos\theta}}\) | \(\displaystyle{\csc\theta=\frac{1}{\sin\theta}}\) |
power reduction (half-angle) formulae |
\(\displaystyle{\sin^2\theta=\frac{1-\cos(2\theta)}{2}}\) | \(\displaystyle{\cos^2\theta=\frac{1+\cos(2\theta)}{2}}\) |
double angle formulae |
\(\sin(2\theta)=2\sin\theta\cos\theta\) | \(\cos(2\theta)=\cos^2\theta-\sin^2\theta\) |
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Triple Integrals in Spherical Coordinates |
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On this page we cover triple integrals in spherical coordinates and several applications. Triple integrals in rectangular coordinates are covered on a separate page, as well as triple integrals in cylindrical coordinates. |
Spherical coordinates are very different from rectangular and cylindrical coordinates. There are still three coordinates, usually labeled \((\rho, \theta, \phi)\), but they are assigned so that spherical-type objects are easy to express. All distances and angles are measured based on a radial line drawn from the origin to the point. Table 1 describes how they are defined and Figure 1 shows this graphically.
Figure 1 (source: Pauls Online Notes: Spherical Coordinates ) |
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\(\rho\) is the radial distance out from the origin to the point |
\(\theta\) is the angle measured in the xy-plane from the positive x-axis to the shadow of the radial line. Note: This is the same angle as \(\theta\) in cylindrical coordinates. |
\(\phi\) is the angle measured from the positive z-axis to the radial line |
Table 1 |
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Note - We have specifically chosen \((\rho, \theta, \phi)\) to describe spherical coordinates but not all books, instructors and videos use these same variables. Some use \((r,\theta,\phi)\) with \(\theta\) and \(\phi\) meaning different things depending on the context. (See the Wiktionary page on spherical coordinates for examples.) Check with your instructor and textbook to see which one they require.
The equations to convert from rectangular to spherical coordinates, in Table 2, are somewhat complicated. So take a few minutes to learn and memorize them (why memorize?). Look for similarities and differences between the equations and see if you can make some sense out of them.
\(x=\rho\sin\phi\cos\theta\) |
\(y=\rho\sin\phi\sin\theta\) |
\(z=\rho\cos\phi\) |
\(\rho^2=x^2+y^2+z^2\) |
Table 2 - \((x,y,z) \to (\rho, \theta, \phi)\) |
Here is a quick video clip discussing these equations and showing a neat animation that will help you get a feel for spherical coordinates.
MIP4U - Introduction to Spherical Coordinates (2min-13secs) | |
Converting from Rectangular to Spherical Coordinates |
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In order to convert from rectangular to spherical coordinates, the equations are Table 2 above are directly applied. The technique is not hard but it will help to watch a couple of quick videos showing how to do this with examples. The first video explains how to convert points from rectangular to spherical. The second video explains how to convert equations from rectangular to spherical. Both videos have plenty of examples. (If you need more examples, check the 17calculus spherical coordinates youtube playlist.)
Krista King Math - Spherical Coordinates (3min-53secs) | |
MIP4U - Converting Between Spherical and Rectangular Equations (8min-15secs) | |
Before we start using spherical coordinates in triple integrals, here is a video that discusses the unit vectors in spherical coordinates.
Michel vanBiezen - Spherical Unit Vector Conversions [9min-45secs] | |
Setting Up and Evaluating Triple Integrals in Spherical Coordinates |
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Okay, so now you know how to convert an equation or a point to spherical coordinates. So how do we set up a triple integral in spherical coordinates? If you are thinking ahead you probably are anticipating that the \(dV\) term has at least one extra term like we had in cylindrical coordinates and you would be right. The \(dV\) term in spherical coordinates has two extra terms, \(\rho^2~\sin\phi\). So \(dV=\rho^2~\sin\phi~d\rho~d\phi~d\theta\). Remember that the order of \(d\rho~d\phi~d\theta\) depends on the order of integration and there are six possible orders. This is just one of them.
In integral form, triple integrals in spherical coordinates look like
\(\displaystyle{ \iiint\limits_V {f(x,y,z) ~ dV} = }\) \(\displaystyle{ \iiint\limits_V{ f(\rho\sin\phi\cos\theta, \rho\sin\phi\sin\theta,\rho\cos\phi ) ~ \rho^2~\sin\phi~d\rho~d\phi~d\theta } }\)
The next step is to describe the volume in spherical coordinates or, in terms of the integral above, determining the limits of integration. The same techniques apply that you used when setting up triple integrals in rectangular coordinates.
Applications of Triple Integrals in Spherical Coordinates |
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Just as with rectangular and cylindrical coordinates, the meaning of the function f will determine what is being calculated with the triple integral. Here is a review of a couple of examples.
f | \(\iiint_V{f~dV}\) |
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1 | volume |
volume density | mass |
Also, physical properties like moment of inertia, center of gravity and force can be calculated using triple integrals.
Okay, time for some practice problems.
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Practice Problems |
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Instructions - Unless otherwise instructed, evaluate these integrals in the spherical coordinate system. (Some of these problems can be successfully worked in more than one coordinate system.)
Level A - Basic |
Practice A01 | |
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Evaluate \(\displaystyle{\iiint\limits_V {x^2+y^2+z^2 ~dV}}\) where V is the unit ball \(x^2+y^2+z^2\leq 1\), using spherical coordinates. | |
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Practice A02 | |
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Integrate \(\displaystyle{f(x,y,z)=3e^{(x^2+y^2+z^2)^{3/2}}}\) over the region inside \(x^2+y^2+z^2=1\) in the first octant, using spherical coordinates. | |
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Practice A03 | |
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Determine the volume outside \(x^2+y^2+z^2=1\) and inside \(x^2+y^2+z^2=4\). | |
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Practice A04 | |
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Calculate the volume of the region outside the cone \(\phi=\pi/3\) and inside the sphere \(\rho=6\cos\phi\). | |
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Practice A05 | |
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Use spherical coordinates to evaluate \(\iiint_V{(x^2+y^2+z^2)^2~dV}\) where V is the ball with center \((0,0,0)\) and radius 5. | |
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Practice A06 | |
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Evaluate, using spherical coordinates, \(\displaystyle{\iiint_V{\frac{1}{x^2+y^2+z^2}dV}}\) where the volume V is between the spheres \(x^2+y^2+z^2=4\) and \(x^2+y^2+z^2=25\) in the first octant. | |
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Level B - Intermediate |