### 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,

the the polar subrectangle and the “polar” volume that the thin box above
recall that, in a one of radius
the length
of an arc subtended by a central angle of
notice that the polar rectangle look at a lot prefer a trapezoid v parallel sides
and also
and also with a width
thus the area that the polar subrectangle is

Simplifying and letting

we have
Therefore, the polar volume that the slim box above ((Figure)) is

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

and
become larger. Hence, we define the polar volume as the limit of the double Riemann sum,

This becomes the expression because that the twin integral.

Definition

The twin integral the the function

end the polar rectangular an ar in the
-plane is defined as

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

as soon as working in polar coordinates. Another way to look in ~ the polar twin integral is to adjust the dual integral in rectangular coordinates by substitution. Once the function
is given in regards to
and
using
changes it to

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),

and