This active learning assignment involves calculating double integrals to summarize:
1. The group members will calculate double integrals over various regions, including rectangles, general regions, and polar coordinates. They will use techniques like iterated integrals and Fubini's theorem.
2. Properties of double integrals like linearity and behavior under transformations will also be explored.
3. Examples will be worked through, such as finding the angle between two planes given their normal vectors, or evaluating a double integral over a specified region.
1. Active Learning Assignment
Subject: Calculus
Topic: Double Integrals
Branch : Electronics & Communication Engineering
Group Members:
Drashti Nakrani (140120111011)
Keerthana Nambiar (140120111012)
Niharika Naruka (140120111013)
Manoj Pandya (140120111014)
2. Double integrals over Rectangle.
Fubini’s Theorem
Properties of double integrals
Double integrals over a general region
Double integrals in polar region
3. f(X1,Y1)δA1+f(X2,Y2)δA2+…..+f(Xn,Yn)δAn = Σ f(Xk,Yk)δAk
.....(1)
Let the number of these sub-regions increase
indefinitely, such that the largest linear dimension
(i.e. diagonal) of δAk approaches zero. The limit of
the sum (1), if it exists, is called the Double
Integral of f(x,y) over the region R and is denoted
by
∫∫ f(x,y) dA.
R
4. Double integrals over a region R may be evaluated in
two successive integrals:
1.Iterated integral
2.Fubini’s theorem
5. Double integrals over a region R may be evaluated by
two successive integrals. In this section in the first part
we see how to express a double integral as an iterated
integral, which can then be evaluated by calculating
two single integrals over the rectangle.
6. Suppose that f is a function of two variables which is
continuous on rectangular region R(where
R=[a, b]x[c, d]) i.e. x=a, x=b, y=c & y=d.
R
x=a
y=d
x=b
y=c
y
x
7. Now if we consider x as constant we can use
This procedure is called partial integration with respect to “y” .
Now,
Is a number that depends on the value of x from x=a to x=b we
can define the function of x as:
A(x)=
d
c
dy
y
x
f )
,
(
b
a
dx
y
x
f )
,
(
d
c
dy
y
x
f )
,
(
8. Now if we integrate the function A w.r.t. x from x=a to
x=b we get,
The integral on the right side of the equation is called
iterated eqn.
dx
dy
y
x
f
dx
x
A
b
a
b
a
d
c
)
,
(
)
(
9. If f is continuous on rectangle
R={(x,y)|a<=x<=b,c<=y<=d)},
then
f(x,y)dA= f(x,y)dydx= f(x,y)dxdy
This is true if we assume that f is bounded on R , f is
discontinuous only on finite number of smooth curves,
integrated integrals exist.
10. Like single integrals, double integrals of continuous
functions have algebraic properties that are useful in
calculations and applications. We assume that all the
following integral exist.
If f(x, y) and g(x, y) are continuous then,
1. Sum and Difference:
D D D
dA
y
x
g
dA
y
x
f
dA
y
x
g
y
x
f )
,
(
)
,
(
)]
,
(
)
,
(
[
11.
D D
dA
y
x
f
c
dA
y
x
cf )
,
(
)
,
(
2. Constant Multiple
3.Domination
a)
b)
( , ) ( , )
R R
f x y dA g x y dA
³
òò òò
If f(x, y) ≥ g(x, y) for all
(x, y) in R, then
R
dA
y
x
f 0
)
,
( 0
)
,
(
y
x
f
If on R
12. 4) Additivity
D D D
dA
y
x
f
dA
y
x
f
dA
y
x
f
1 2
)
,
(
)
,
(
)
,
(
If D=D1 U D2, where
regions R1 and R2 do not
overlap perhaps on their
boundries as shown in fig.
19.
D
)
(
h
)
(
h
2
1
R
b
a
2
1
)rdrd
rsin
,
f(rcos
y)dA
f(x,
then
D
on
continuous
is
f
If
region.
polor
a
be
)}
(
h
r
)
(
h
,
|
)
{(r,
D
Let
2.
)rdrd
rsin
,
f(rcos
y)dA
f(x,
then
R,
on
continuous
is
f
If
2
-
0
and
rectangle
polar
a
be
}
b,
r
a
|
)
{(r,
R
Let
1.
Properties
