For some regions R, it convenient to convert to polar coordinates in order to evaluate the double integral
Consider the sector a<=r<=b, c<=theta<=d shown in the figure below.
Recall that x=r*cos(theta) and y=r*sin(theta). The double integral is given by:
In the above formula one integrates with respect to theta first, then r. Alternatively, one could integrate with respect to r first, then theta.
Discussion of the Iterated Integral in Polar Coordinates
In the case of double integral in polar coordinates we made the connection dA=dxdy. dxdy is the area of an infinitesimal rectangle between x and x+dx and y and y+dy. In polar coordinates, dA=rd(theta)dr is the area of an infinitesimal sector between r and r+dr and theta and theta+d(theta). See the figure below.
The area of the region is the product of the length of the region in theta direction and the width in the r direction. The width is dr. The length is d(theta)*r, the arclength of a part of a circle of angle d(theta). (The radius is essentially constant in the region since dr is infinitesimal.)
Example
Consider the integral with f(x,y)=2x+3y^2 where R is the region between the circles x^2+y^2=1 and x^2+y^2=4. In this case 1<=r<=2 and 0<=theta<=2*pi. We can convert the function f(x,y) into polar coordinates with the substitutions x=r*cos(theta) and y=r*sin(theta). The iterated integral is
We integrate with respect to theta first, then r. Alternatively, we have
Examples of how to evaluate iterated integrals are given in the double integrals page.
General Regions
If the region R is of the form g_1(r)<=theta<=g_2(r) with a<=r<=b, as shown in the figure below,
then the double integral is given by the iterated integral
If the region R is of the form h_1(theta)<=r<=h_2(theta) with c<=theta<=d, as shown in the figure below,
then the double integral is given by the iterated integral
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