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Double Integrals in Polar Coordinates |
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This page covers double integrals in polar coordinates. Double integrals in rectangular coordinates are covered on a separate page. |
As you learned on the polar coordinates page, you use the equations \(x=r\cos\theta\) and \(y=r\sin\theta\) to convert equations from rectangular to polar coordinates. The same idea applies to a function and the description of an area in the xy-plane.
For example, if you have an integral in rectangular coordinates that looks like \(\iint_A{f(x,y)~dA}\), you need to do three things to convert this to polar coordinates.
1. First, substitute for x and y in \(f(x,y)\) using the above equations to get \(f(r\cos\theta,r\sin\theta)\). This new form of the function can be written as \(f(r,\theta)\).
2. Second, describe the area in polar coordinates or, if the area is already given in rectangular coordinates, convert the area in the xy-plane from rectangular to polar coordinates.
3. Finally, set up the integral with the function \(f(r,\theta)\) in polar coordinates, being careful to integrate in the correct order.
Okay, so that is the big picture but how do you implement this when working problems? Time for some videos. Both of these videos are rather long but they will give you a good handle on double integrals in polar coordinates. You do not need to watch both of them since they cover the same ideas. But if you do, you will have a better understanding of the techniques.
MIT OCW - Lec 17 | MIT 18.02 Multivariable Calculus, Fall 2007 (51min-29secs) | |
Evans Lawrence - Multivariable Calculus: Lecture 19 - Double Integration in Polar Coordinates (35min-2secs) | |
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Practice Problems |
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Instructions - - Unless otherwise instructed, evaluate the following integrals using polar coordinates, giving your answers in exact terms.
Level A - Basic |
Practice A01 | |
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\(\displaystyle{\int_{0}^{1/\sqrt{2}}{ \int_{y}^{\sqrt{1-y^2}}{ 3y~dx~dy} } }\) | |
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Practice A02 | |
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\(\displaystyle{\int_{0}^{2}{ \int_{-\sqrt{2y-y^2}}^{\sqrt{2y-y^2}}{\sqrt{x^2+y^2}~dx~dy}}}\) | |
answer |
solution |
Practice A03 | |
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Determine the volume of the solid below the surface \(f(x,y)=4-x^2-y^2\) above the xy-plane over the region bounded by \(x^2+y^2=1\) and \(x^2+y^2=4\). | |
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Practice A04 | |
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Determine the volume of \(z=\sqrt{9-x^2-y^2}\) over the region \(x^2+y^2\leq 4\) in the first octant. | |
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Practice A05 | |
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Evaluate \(\iint_{A}{e^{-x^2-y^2}~dA}\) where A is bounded by \(x=\sqrt{4-y^2}\) and the y-axis. | |
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solution |
Practice A06 | |
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\(\displaystyle{\int_{x=1}^{2}{\int_{y=0}^{x}{\frac{1}{(x^2+y^2)^{3/2}}~dy~dx}}}\) | |
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Practice A07 | |
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Convert the integral \(\displaystyle{ \int_{x=0}^{1}{ \int_{y=x^2}^{x}{ f~dy~dx }}}\) to polar coordinates. | |
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Practice A08 | |
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Convert \(\displaystyle{ \int_{y=0}^{2}{\int_{x=0}^{\sqrt{2y-y^2}}{f~dx~dy}}}\) to polar coordinates. | |
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Practice A09 | |
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\(\displaystyle{\int_{0}^{3}{ \int_{0}^{\sqrt{9-x^2}}{\sqrt{(x^2+y^2)^3}~dy~dx}}}\) | |
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Practice A10 | |
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\(\displaystyle{\int_{-3}^{3}{\int_{0}^{\sqrt{9-x^2}}{\sin(x^2+y^2)~dy~dx}}}\) | |
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Practice A11 | |
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Use a double polar integral to find the volume of the solid enclosed by \(-x^2-y^2+z^2=1\) and \(z=2\). | |
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solution |
Practice A12 | |
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A cylindrical drill with a radius of 5 cm is used to bore a hole through the center of a sphere of radius 7 cm. Find the volume of the ring shaped solid that remains. | |
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Level B - Intermediate |
Practice B01 | |
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Calculate the area under the plane \(6x+4y+z=12\) above the disk with boundary circle \(x^2+y^2=2y\). | |
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Practice B03 | |
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Find the surface area of the part of the plane \(z=3+2x+4y\) that lies inside the cylinder \(x^2+y^2=4\). | |
answer |
solution |