Double Integrals in Polar Coordinates

    Some double integrals are much easier to evaluate in polar form than in rectangular form. This is especially true for regions such as circles, cardioids, and rose curves, and for integrands that  involve a sum of squares. In this topic we illustrate how to convert a Cartesian integral into polar integral; in particular, we emphasis sketching the region of integration.

    Polar coordinates are used in double integrals primarily when the integrand or the region of integration (or both) have relatively simple polar descriptions. The polar conversion formulas are used to convert from rectangular to polar coordinates:  

Proposition (Double Integral in Polar Coordinates) If is continuous in the polar region described by and (with ), then



Thus the procedure for changing from a Cartesian integral into a polar integral requires substitution of



into the Cartesian integral and then converting the region of integration to polar form  

Example (Double Integral in Polar Coordinates) Use polar coordinates to evaluate the integrals.

(a) Evaluate where is the region in the upper half-plane bounded by the circles and
    
    Solution. The region is described as



It is an upper ring with polar coordinates given by  




Therefore,  









(b) Find the volume of the solid bounded by the plane and the paraboloid

    Solution. If we put in the equation of the paraboloid, we get This means that the plane intersects the paraboloid in the circle so the solid lies under the paraboloid and above the circular disk given by In polar coordinates is given by



Since the volume is

Example (Evaluating a Double Integral in Polar Coordinates) Use polar coordinates to evaluate the integrals.

(a) Use a double integral to find the area enclosed by one loop of the four leaved rose
    
    Solution. From the sketch of the curve we see that a loop is given by the region





So the area is  



(b) Find the volume of the solid that lies under the paraboloid above the and inside the cylinder

    Solution. The solid lies above the disk whose boundary circle has equation or In polar coordinates the boundary is or Thus the disk is given by





and so the volume is

Example (Using Double Integrals in Polar Coordinates) Use polar coordinates to evaluate the integrals.

(a) Sketch the region and evaluate the double integral given

    Solution. We have



and the region of integration is



(b) Sketch the region of integration and evaluate the double integral given   

    Solution. We have



and the region of integration is



(c) Sketch the region of integration and evaluate the double integral given

    Solution. We have



and the region of integration is

Example (Determining a Double Integral in Polar Coordinates) Use polar coordinates to evaluate the integrals.

(a) Sketch the region of integration and evaluate the double integral given

    Solution. We have



and the region of integration is



(b) Sketch the region of integration and evaluate the double integral given .

    Solution. We have



and the region of integration is

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Cite this as:
Double Integrals In Polar Coordinates
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/double-integrals-in-polar-coordinates.html