Please enable JavaScript.
Coggle requires JavaScript to display documents.
Integral calculus, multiple integrals, Difficult integration,…
Integral calculus
riemann integral
sequential defination bounded function closed bounded interval
lower RIEMANN sum = upper RIEMANN sum closed interval bounded function
Vs riemann sum
Bounded necessary??yes
Note every bounded function is not riemann integrable
specific
monotone function closed and bounded interval is riemannn integrable
Ask if not closed and bounded interval
continous function over closed and bounded interval
FINITLELT MANY DISCONTINIUTY AND BOUNDED FUNCTION
questions
why we deifne on closed bounded interval
why f bounded what if not ???
Properties(very similar to integration i. 12 th)
Sum of two f, g
Multiple f by c
f<g then
|int f | < int|f|
Splitting the integral
Partition of a interval
Set of points
Refinement of partition
Given partition make a new partition which has all the points in the original partition. And some extra points aka. old partition is a proper subset of the new
norm of refinement is lesser
CAUTION norm of A < NORM B does not imply that a in the refienemnet of B
typical partitions
i/in
GP a gp like 1 ,r,,r2,,r3,,,,,,,rn
TILL NOW ALL F HAS TO BOUNDED
in terms of riemann sum and eplision
limit involing upper and lower sums
RIEMANN INTEGRATION THEORY CLOSED BOUNDED INTERVAL AND F IS BOUNDED FUNCTION INITALLY
Proving is or is not
Use of inequalities
Disprove sin1/x
Prove xsqaure
Sandwich
Improper intrgrak
Type1
One of the end point is +_infinity
Used the limit of that end point
Exist and finite converge
Otherwise does not converge
Test convegence
Comparison test
Both integrands must be non negative in the required interval
Same conclusions as in series
Only single implicqtion
Limit comparison test
Both positive function in that domain
Trick if both negative then test the convergence of their negative cpunterparts
Limit find
Zero
Finite
Inifinity
Dirchlet test
One function is bounded and montonic on starting to infinity
Limit at infinity must be zero
Other functions integral is bounded
Usefyl in sin cos where bounded integrals
Absolutely convergence
USE IT AS A TEST
IF SOMETHING IS ABSOLUTELY CONVERGENT THEN IT IMPLIES CONVERGENCE
Type 2
Infinite. Discontinuity at a point in between or end point of the limits DOUBT only infinite discontinuity?????
Be careful to check the discontinuous points in between the interval
Both type 1 and type 2
Break into two integrals one of each type
Be careful sometimes it looks infinite discontinuity but we have to check that by finding the limit
Many times break the integral so as to apply the tests
Example the function may be continous on some part of domain
If substituting do indefinite don't keep chaniiging the limits
Eplison greater than zero means that limit eplision tends to zero plus
Not converge vs diverge
IMPROPER INTGARAL
multiple integrals
important skills
Analyze Domain
plot it
transform it using the change of variables
will it always be rectangular coordinates ????????
focus not on transforming equations but on the edgese of the actual region in cartesian coordinate
why
nicer domain
simplify the integral
change the order of integration and its limits
equation of cone 3D
cone z = sqrt(x2+y2)
equation cardidoid
equation of cylinder
x2 + y2 = constant
parabloid
z= x2 +y2
intersection of planes
tetrahedron
Change of variable OR transformation
use of Jacobian
what is it
gives the appropriate factor
don't forget the modulus
typical
polar
spherical
cylindrical
how to decide which transformation is useful
use symmetry
double integrals
function of two variables
because of that domain is now a area or surface
APPLICATION
finding the area between two curves
finding the volume of solid
evaluate Beta Gamma function s
finding gamma half
finding the relation between beta and gamma functions
evaluate some single variable integrals
Area of domain
If fxy is separable into gx and hy and rectangular doamin
triple integrals
function of three variables
so Domain is volume in 3D space
APPPLICATION
finding the volume of solid
typical
described by rotating a curve along an axis
finding
Difficult integration
Change order
xregular y regular
Multiple integrals
Square the integral
Then write in two coordinates same integral
Convert to multiple integral
Change coordinates
Multivariable differential calculus
limits
polar coordinates defination
sequential defination
Epsilion delta defination
diffrentiablility
not differentiable
assume equality and rearranege terms and show that when we take limit
Calculus
First fundamental theorem of calculus
Second fundamental theorem of calculus