Learning Objectives
Recognize the layout of a twin integral end a polar rectangle-shaped region.Evaluate a double integral in polar coordinates by making use of an iterated integral.Recognize the layout of a dual integral end a general polar region.Use dual integrals in polar coordinates to calculation areas and also volumes.You are watching: Use a double integral to find the area of the region. one loop of the rose r = 4 cos(3θ)
Double integrals are periodically much simpler to evaluate if we adjust rectangular collaborates to polar coordinates. However, prior to we define how to make this change, we require to establish the concept of a double integral in a polar rectangular region.
Polar rectangular Regions of Integration
When we characterized the double integral for a continuous function in rectangular coordinates—say,










Simplifying and letting



Using the very same idea for all the subrectangles and also summing the quantities of the rectangle-shaped boxes, we attain a double Riemann amount as
As we have seen before, we obtain a better approximation come the polar volume that the solid over the an ar once we allow


This becomes the expression because that the twin integral.
Definition
The twin integral the the function


Again, just as in double Integrals over rectangle-shaped Regions, the double integral end a polar rectangular an ar can be expressed as an iterated integral in polar coordinates. Hence,
Notice that the expression for is replaced by





Note the all the properties provided in twin Integrals over rectangle-shaped Regions because that the double integral in rectangular collaborates hold true for the twin integral in polar coordinates as well, so we deserve to use them without hesitation.
As we deserve to see from (Figure),




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Now the we have actually sketched a polar rectangle-shaped region, permit us show how to advice a twin integral over this region by using polar coordinates.
First we lay out a figure comparable to (Figure) yet with outer radius
