Multiple Integrals
Double Integrals over rectangles
and general regions- Double
integrals in Polar coordinates
Here is our Team
(Group-D)
(Technical Team)
BHAVYA​BALCHANDANI​23BCE10151
SHIVANG​SHAH​23BCE10158​
PARTH ​JAIN​23BCE10156​
​VANSH ​TIWARI​23BCE10136​
Here is our Team
(Group-D)
(Non-Technical Team)
ARNAV​SHUKLA​23BCE10173​
SHLOK​PASTAGIA 23BCE10190
DHRUV ​KUMAR​​23BCE10160
TANPREET​SINGH​​23BCE10159​
SAMRIDDHI ​TRIPATHI​23BCE10140​
​VANSH ​TIWARI​23BCE10136​
The forthcoming presentation
will go into the following topics
1. Introduction
2. Double Integrals and It’s Properties
3. Double Integrals over rectangles
4. Double Integrals over general regions
5. Double integrals in Polar coordinates
6. Conclusion
Introduction
Double integrals are a fundamental concept in
calculus for calculating quantities such as area,
volume, or average values of functions over twodimensional regions .
Double integral is defined as the integrals of a
function in two variables over a region in R, i.e.
the real number plane.
The double integral of a function of two
variables, say f(x ,y) over a region can be
denoted as: R f ( x ,y ) dA = R f ( x ,y ) dx dy
∬
∬
Properties of Double
Integrals
The properties of double integrals are as
follows:
∫x=ab ∫y=cd f(x,y)dy.dx = ∫y=cd∫x=ab f(x,y)dx.dy
∫∫(f(x,y) ± g(x,y)) dA = ∫∫f(x,y)dA ± ∫∫g(x,y)dA
If f(x,y) < g(x,y), then ∫∫f(x,y)dA < ∫∫g(x,y)dA
k ∫∫f(x,y).dA = ∫∫k.f(x,y).dA
∫∫R Sf(x,y).dA = ∫∫Rf(x,y).dA+∫∫sf(x,y).dA
∪
Example of Double Integrals
Double Integrals over rectangles
Double Integrals over Rectangles:
1. Rectangular Region: When working with a rectangular region in the xyplane, the process is relatively straightforward.
2. Iterated Integral: Double integrals over rectangles are typically computed
using an iterated integral, where one variable (usually x or y) is integrated
with respect to the other held constant.
3. Divide and Conquer: You break down the rectangular region into smaller
rectangles and evaluate the integral over each of these rectangles. Then,
you sum up the contributions from all these rectangles.
4. Fubini's Theorem: Fubini's theorem states that the order of integration
(whether integrating with respect to x first or y first) can be interchanged as
long as the function is well-behaved.
Example of Double Integrals over rectangles
Example of Double Integrals over rectangles
Double Integrals over general regions
Double Integrals over General Regions:
1. Arbitrary Shapes: General regions can have irregular and more complex shapes and
boundaries that are not easily divided into rectangles.
2. Coordinate Transformation: In these cases, you often need to use coordinate
transformations to change to a coordinate system where the region becomes
simpler (e.g., polar coordinates for circular regions).
3. Integration over Transformed Region: After the coordinate transformation, you set
up the double integral over the transformed region and integrate. This may involve
iterated integrals, where both variables are functions of new coordinates.
4. Versatility: Double integrals over general regions are more versatile and can be
used to calculate quantities over a wide range of irregular shapes, making them a
valuable tool in solving real-world problems.
Example of Double Integrals over general regions
Example of Double Integrals over
general regions
Double integrals in Polar coordinates
1. Conversion to Polar Coordinates:
Instead of using the traditional rectangular coordinates (x, y), you
convert the problem to polar coordinates (r, θ).
In polar coordinates, r represents the radial distance from the origin,
and θ represents the angle between the positive x-axis and the line
connecting the origin to the point of interest.
2. Change of Variables:
You replace dx dy with r dr dθ in the double integral, essentially
changing the variables of integration.
This substitution is necessary because the differential area element
in polar coordinates is r dr dθ.
3.Limits of Integration:
You also need to change the limits of integration. The limits for r and θ
depend on the specific region you're integrating over.
For r, you typically use limits that correspond to the radial distance within
the region.
For θ, you choose limits based on the angular extent of the region.
4.Function in Polar Form:
You express the function you're integrating in terms of r and θ. This may
involve rewriting the function to fit polar coordinates.
5.Evaluate the Integral:
With the integral set up in polar coordinates and the correct limits of
integration, you can proceed to calculate the double integral.
This often involves breaking the region into smaller pieces and summing
their contributions, just as in rectangular coordinates
Example of Double integrals in Polar
coordinates
Conclusion
In summary, double integrals over rectangles are suitable for regular
shapes and are computed by summing contributions from smaller
rectangles. In contrast, double integrals over general regions are applied
to irregular shapes and may involve coordinate transformations to
simplify the integration process. Both approaches are essential tools in
calculus for solving a variety of problems related to areas, volumes, and
averages
Double integrals in polar coordinates are particularly useful when dealing
with problems that have circular or annular symmetry, like finding areas,
moments of inertia, or mass distributions in circular regions. By using
polar coordinates, you can simplify the calculations and make them more
intuitive for such problems .
References
We have taken inspiration for our presentation from Thomas Calculus Book
Notes of Prof. M. Annamalai
Engineering Mathematics book
Thank You