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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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Pauls Online Notes: Triple Integrals in Cylindrical Coordinates |
Triple Integrals in Cylindrical Coordinates |
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On this page we cover triple integrals in cylindrical coordinates and several applications. Triple integrals in rectangular coordinates are covered on a separate page, as well as triple integrals in spherical coordinates. |
When you studied polar coordinates, you learned that you could take an area in the plane in rectangular coordinates and use polar coordinates to describe the same area. To do this, you used the equations \(x=r\cos(\theta)\) and \(y=r\sin(\theta)\). These equations convert the equations for an area in the plane from rectangular coordinates \((x,y)\) to polar coordinates \((r,\theta)\).
However, now we have three dimensions. In rectangular coordinates, we have \((x,y,z)\). One of the nice things about cylindrical coordinates is that we use the same equations on x and y to get r and \(\theta\) and to go to cylindrical coordinates z does not change. Another way of looking at it is that we take polar coordinates \((r,\theta)\) and slap on the rectangular coordinate z to the end to get \((r,\theta,z)\) and call this cylindrical coordinates. So the cylindrical coordinates conversion equations are given in Table 1 and Figure 1 shows this relationship.
\(x=r\cos(\theta)\) |
\(y=r\sin(\theta)\) |
\(z=z\) |
\(r^2=x^2+y^2\) |
Table 1 |
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Figure 1 [ source: Paul's Online Notes - Cylindrical Coordinates ] |
Figure 2 [ source: Wikiversity - Cylindrical Coordinates ] |
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Notation Note - Figure 2 shows another way to describe cylindrical coordinates. Some books, instructors, videos and sites use \((\rho,\phi,z)\) to describe the same point as \((r,\theta,z)\). We choose to use \((r,\theta,z)\) for cylindrical coordinates since the cylindrical coordinate system is so closely related to two-dimensional polar coordinates usually described at \((r,\theta)\). As always, check with your instructor to see what they expect you to use.
Also in Figure 2, they are calling A the polar axis (positive x-axis) and L is the positive z-axis.
Describing Surfaces and Volumes in Cylindrical Coordinates |
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So, you may ask, why do we need the cylindrical coordinate system when we describe surfaces and volumes in the rectangular coordinate system? Well, there are two reasons.
1. Some surfaces and volumes are more easily (simply) described in cylindrical coordinates. An example is given below.
2. When we get to triple integrals, some integrals are more easily evaluated in cylindrical coordinates and you will even have some integrals that can't be evaluated in rectangular coordinates but can be in cylindrical.
Setting Up and Evaluating Triple Integrals in Cylindrical Coordinates |
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Let's start by watching this short video clip, explaining cylindrical coordinates again and showing how to set up triple integrals in cylindrical coordinates.
MIP4U - Triple Integrals Using Cylindrical Coordinates (2min-7secs) | |
A couple of clarifications are in order that he mentions in this video.
First, he mentions that there is an extra factor of r in the term \(dV\) in cylindrical coordinates but he doesn't explain where it comes from. Remember from your study of polar coordinates that \(dA = dx~dy\) in rectangular coordinates becomes \(dA = r~dr~d\theta\). This next video clip explains where \(dV\) comes from in more detail.
Larson Calculus - Triple Integrals in Cylindrical Coordinates (5min-26secs) | |
The second clarification involves the order of components of \(dV\). In the first video, he says that \(dV = r~dz~dr~d\theta\). This is only one of six possible representations for \(dV\). The order is determined by how the volume is described by the equations. To determine what order to integrate when setting up the integral, we use the same idea as we did when setting up double integrals.
In integral form, a triple integral in cylindrical coordinates looks like
\(\displaystyle{ \iiint\limits_V {f(x,y,z) ~ dV} = }\) \(\displaystyle{ \iiint\limits_V{ f(r\cos\theta, r\sin\theta,z)~r~dr~d\theta~dz } }\)
Before going on to spherical coordinates, you may want to work some practice problems involving cylindrical coordinates.
next: triple integrals (spherical) → |
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Practice Problems |
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Instructions - Unless otherwise instructed, evaluate these integrals in cylindrical coordinates.
Level A - Basic |
Practice A01 | |
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Integrate \(f(x,y,z)=x^2+y^2\) over the solid region bounded by \(z=0\) and \(z=4-\sqrt{x^2+y^2}\) in cylindrical coordinates. | |
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Practice A02 | |
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\(\displaystyle{\int_{-2}^{2}{\int_{-\sqrt{4-y^2}}^{\sqrt{4-y^2}}{\int_{\sqrt{x^2+y^2}}^{2}{xz~dz~dx~dy}}}}\) | |
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Practice A03 | |
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Calculate the volume of the solid bounded by \(z=0\) and \(z=9-x^2-y^2\). | |
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Practice A04 | |
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Calculate the volume of the solid bounded by \(z=4-\sqrt{x^2+y^2}\), \(x^2+y^2=1\) and \(z=0\). | |
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Practice A05 | |
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Evaluate \(\iiint_{V}{x^2~dV}\) where V is the solid that lies within the cylinder \(x^2+y^2=1\) above the plane \(z=0\) and below the cone \(z^2=4x^2+4y^2\). | |
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Practice A06 | |
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\(\displaystyle{\int_{-1}^{1}{\int_{-\sqrt{1-x^2}}^{\sqrt{1-x^2}}{\int_{0}^{2}{\sqrt{x^2+y^2}~dz~dy~dx}}}}\) | |
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Practice A07 | |
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\(\displaystyle{\int_{-2}^{2}{\int_{-\sqrt{4-x^2}}^{\sqrt{4-x^2}}{\int_{x^2+y^2}^{4}{y~dz~dy~dx}}}}\) | |
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solution |
Level B - Intermediate |
Practice B01 | |
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Integrate \(f(x,y,z)=1\) over the solid region bounded by \(z=0\), \(x^2+y^2=1\), \(x^2+y^2=4\) and \(z=4-(x^2+y^2)\) in cylindrical coordinates. | |
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Practice B02 | |
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Express as a triple integral, using cylindrical coordinates, the volume of the region above the cone \(z=\sqrt{x^2+y^2}\) and inside the sphere \(x^2+y^2+z^2=2az, a >0\). | |
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Practice B03 | |
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Use cylindrical coordinates to find the volume outside the cylinder \(x^2+y^2=1\), (\(z \ge 0\)) and inside the paraboloid \(z=4-x^2-y^2\). | |
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