# 5.3 Double integrals in polar coordinates

 Page 1 / 7
• Recognize the format of a double integral over a polar rectangular region.
• Evaluate a double integral in polar coordinates by using an iterated integral.
• Recognize the format of a double integral over a general polar region.
• Use double integrals in polar coordinates to calculate areas and volumes.

Double integrals are sometimes much easier to evaluate if we change rectangular coordinates to polar coordinates. However, before we describe how to make this change, we need to establish the concept of a double integral in a polar rectangular region.

## Polar rectangular regions of integration

When we defined the double integral for a continuous function in rectangular coordinates—say, $g$ over a region $R$ in the $xy$ -plane—we divided $R$ into subrectangles with sides parallel to the coordinate axes. These sides have either constant $x$ -values and/or constant $y$ -values. In polar coordinates, the shape we work with is a polar rectangle    , whose sides have constant $r$ -values and/or constant $\theta$ -values. This means we can describe a polar rectangle as in [link] (a), with $R=\left\{\left(r,\theta \right)|a\le r\le b,\alpha \le \theta \le \beta \right\}.$

In this section, we are looking to integrate over polar rectangles. Consider a function $f\left(r,\theta \right)$ over a polar rectangle $R.$ We divide the interval $\left[a,b\right]$ into $m$ subintervals $\left[{r}_{i-1},{r}_{i}\right]$ of length $\text{Δ}r=\left(b-a\right)\text{/}m$ and divide the interval $\left[\alpha ,\beta \right]$ into $n$ subintervals $\left[{\theta }_{i-1},{\theta }_{i}\right]$ of width $\text{Δ}\theta =\left(\beta -\alpha \right)\text{/}n.$ This means that the circles $r={r}_{i}$ and rays $\theta ={\theta }_{i}$ for $1\le i\le m$ and $1\le j\le n$ divide the polar rectangle $R$ into smaller polar subrectangles ${R}_{ij}$ ( [link] (b)).

As before, we need to find the area $\text{Δ}A$ of the polar subrectangle ${R}_{ij}$ and the “polar” volume of the thin box above ${R}_{ij}.$ Recall that, in a circle of radius $r,$ the length $s$ of an arc subtended by a central angle of $\theta$ radians is $s=r\theta .$ Notice that the polar rectangle ${R}_{ij}$ looks a lot like a trapezoid with parallel sides ${r}_{i-1}\text{Δ}\theta$ and ${r}_{i}\text{Δ}\theta$ and with a width $\text{Δ}r.$ Hence the area of the polar subrectangle ${R}_{ij}$ is

$\text{Δ}A=\frac{1}{2}\text{Δ}r\left({r}_{i-1}\text{Δ}\theta +{r}_{1}\text{Δ}\theta \right).$

Simplifying and letting ${r}_{ij}^{*}=\frac{1}{2}\left({r}_{i-1}+{r}_{i}\right),$ we have $\text{Δ}A={r}_{ij}^{*}\text{Δ}r\text{Δ}\theta .$ Therefore, the polar volume of the thin box above ${R}_{ij}$ ( [link] ) is

$f\left({r}_{ij}^{*},{\theta }_{ij}^{*}\right)\text{Δ}A=f\left({r}_{ij}^{*},{\theta }_{ij}^{*}\right){r}_{ij}^{*}\text{Δ}r\text{Δ}\theta .$

Using the same idea for all the subrectangles and summing the volumes of the rectangular boxes, we obtain a double Riemann sum as

$\sum _{i=1}^{m}\sum _{j=1}^{n}f\left({r}_{ij}^{*},{\theta }_{ij}^{*}\right){r}_{ij}^{*}\text{Δ}r\text{Δ}\theta .$

As we have seen before, we obtain a better approximation to the polar volume of the solid above the region $R$ when we let $m$ and $n$ become larger. Hence, we define the polar volume as the limit of the double Riemann sum,

$V=\underset{m,n\to \infty }{\text{lim}}\sum _{i=1}^{m}\sum _{j=1}^{n}f\left({r}_{ij}^{*},{\theta }_{ij}^{*}\right){r}_{ij}^{*}\text{Δ}r\text{Δ}\theta .$

This becomes the expression for the double integral.

## Definition

The double integral of the function $f\left(r,\theta \right)$ over the polar rectangular region $R$ in the $r\theta$ -plane is defined as

$\underset{R}{\iint }f\left(r,\theta \right)dA=\underset{m,n\to \infty }{\text{lim}}\sum _{i=1}^{m}\sum _{j=1}^{n}f\left({r}_{ij}^{*},{\theta }_{ij}^{*}\right)\text{Δ}A=\underset{m,n\to \infty }{\text{lim}}\sum _{i=1}^{m}\sum _{j=1}^{n}f\left({r}_{ij}^{*},{\theta }_{ij}^{*}\right){r}_{ij}^{*}\text{Δ}r\text{Δ}\theta .$

Again, just as in Double Integrals over Rectangular Regions , the double integral over a polar rectangular region can be expressed as an iterated integral in polar coordinates. Hence,

$\underset{R}{\iint }f\left(r,\theta \right)dA=\underset{R}{\iint }f\left(r,\theta \right)r\phantom{\rule{0.2em}{0ex}}dr\phantom{\rule{0.2em}{0ex}}d\theta =\underset{\theta =\alpha }{\overset{\theta =\beta }{\int }}\phantom{\rule{0.2em}{0ex}}\underset{r=a}{\overset{r=b}{\int }}f\left(r,\theta \right)r\phantom{\rule{0.2em}{0ex}}dr\phantom{\rule{0.2em}{0ex}}d\theta .$

where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
can you provide the details of the parametric equations for the lines that defince doubly-ruled surfeces (huperbolids of one sheet and hyperbolic paraboloid). Can you explain each of the variables in the equations?