Learning Objectives

Recognize the layout of a double integral over a polar rectangle-shaped region.Evaluate a double integral in polar works with by making use of an iterated integral.Recognize the style of a dual integral end a general polar region.Use twin integrals in polar coordinates to calculation areas and volumes.

You are watching: For the region r below, write ∬rfda as an iterated integral in polar coordinates.


Double integrals are periodically much easier to evaluate if we change rectangular coordinates to polar coordinates. However, prior to we define how to make this change, we require to establish the principle of a dual integral in a polar rectangle-shaped region.


Polar rectangle-shaped Regions that Integration

When we characterized the dual integral for a continuous function in rectangular coordinates—say,

*
that the polar subrectangle and also the “polar” volume the the thin box over
*
remind that, in a circle of radius
*
the length
*
of an arc subtended through a main angle of
*
radians is
*
an alert that the polar rectangle watch a lot like a trapezoid with parallel political parties
*
and also
*
and also with a width
*
therefore the area the the polar subrectangle is


*

Simplifying and also letting

*
we have actually
*
Therefore, the polar volume that the thin box over ((Figure)) is


*

Using the exact same idea for every the subrectangles and also summing the volumes of the rectangular boxes, we acquire a twin Riemann amount as


As we have actually seen before, we acquire a better approximation come the polar volume of the solid over the an ar once we permit

*
and also
*
come to be larger. Hence, we specify the polar volume together the border of the double Riemann sum,


This i do not care the expression because that the double integral.


Definition

The double integral of the function

*
over the polar rectangular region in the
*
-plane is characterized as


Again, simply as in twin Integrals over rectangle-shaped Regions, the twin integral over a polar rectangular region can it is in expressed as an iterated integral in polar coordinates. Hence,


Notice that the expression for is replaced by

*
when working in polar coordinates. Another method to look in ~ the polar double integral is to readjust the double integral in rectangular collaborates by substitution. As soon as the role
*
is provided in terms of
*
and
*
utilizing
*
changes it to


Note the all the properties provided in twin Integrals over rectangle-shaped Regions for the twin integral in rectangular coordinates hold true for the dual integral in polar collaborates as well, for this reason we deserve to use them without hesitation.


As we have the right to see native (Figure),

*
and also
*
space circles the radius
*
and also
*
covers the entire top fifty percent of the plane. Thus the region looks prefer a semicircular band.

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Now that we have actually sketched a polar rectangle-shaped region, allow us show how to evaluate a dual integral end this an ar by making use of polar coordinates.


First we map out a figure similar to (Figure) but with outer radius

*
indigenous the figure we have the right to see that we have