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Double Integrals in Polar Coordinates 

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 xyplane.
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 xyplane 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 (51min29secs)  
Evans Lawrence  Multivariable Calculus: Lecture 19  Double Integration in Polar Coordinates (35min2secs)  
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Practice Problems 

Instructions   Unless otherwise instructed, evaluate the following integrals using polar coordinates, giving your answers in exact terms.
Level A  Basic 
Practice A01  

\(\displaystyle{\int_{0}^{1/\sqrt{2}}{ \int_{y}^{\sqrt{1y^2}}{ 3y~dx~dy} } }\)  
answer 
solution 
Practice A02  

\(\displaystyle{\int_{0}^{2}{ \int_{\sqrt{2yy^2}}^{\sqrt{2yy^2}}{\sqrt{x^2+y^2}~dx~dy}}}\)  
answer 
solution 
Practice A03  

Determine the volume of the solid below the surface \(f(x,y)=4x^2y^2\) above the xyplane over the region bounded by \(x^2+y^2=1\) and \(x^2+y^2=4\).  
answer 
solution 
Practice A04  

Determine the volume of \(z=\sqrt{9x^2y^2}\) over the region \(x^2+y^2\leq 4\) in the first octant.  
answer 
solution 
Practice A05  

Evaluate \(\iint_{A}{e^{x^2y^2}~dA}\) where A is bounded by \(x=\sqrt{4y^2}\) and the yaxis.  
answer 
solution 
Practice A06  

\(\displaystyle{\int_{x=1}^{2}{\int_{y=0}^{x}{\frac{1}{(x^2+y^2)^{3/2}}~dy~dx}}}\)  
answer 
solution 
Practice A07  

Convert the integral \(\displaystyle{ \int_{x=0}^{1}{ \int_{y=x^2}^{x}{ f~dy~dx }}}\) to polar coordinates.  
answer 
solution 
Practice A08  

Convert \(\displaystyle{ \int_{y=0}^{2}{\int_{x=0}^{\sqrt{2yy^2}}{f~dx~dy}}}\) to polar coordinates.  
answer 
solution 
Practice A09  

\(\displaystyle{\int_{0}^{3}{ \int_{0}^{\sqrt{9x^2}}{\sqrt{(x^2+y^2)^3}~dy~dx}}}\)  
answer 
solution 
Practice A10  

\(\displaystyle{\int_{3}^{3}{\int_{0}^{\sqrt{9x^2}}{\sin(x^2+y^2)~dy~dx}}}\)  
answer 
solution 
Practice A11  

Use a double polar integral to find the volume of the solid enclosed by \(x^2y^2+z^2=1\) and \(z=2\).  
answer 
solution 
Practice A12  

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.  
answer 
solution 
Level B  Intermediate 
Practice B01  

Calculate the area under the plane \(6x+4y+z=12\) above the disk with boundary circle \(x^2+y^2=2y\).  
answer 
solution 
Practice B03  

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 