Integral calculus

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

riemann integral

sequential defination bounded function closed bounded interval

lower RIEMANN sum = upper RIEMANN sum closed interval bounded function

specific

questions

why we deifne on closed bounded interval

why f bounded what if not ???

monotone function closed and bounded interval is riemannn integrable

continous function over closed and bounded interval

Ask if not closed and bounded interval

Vs riemann sum

Properties(very similar to integration i. 12 th)

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

Sum of two f, g

Multiple f by c

f<g then

|int f | < int|f|

Splitting the integral

Bounded necessary??yes

Note every bounded function is not riemann integrable

TILL NOW ALL F HAS TO BOUNDED

FINITLELT MANY DISCONTINIUTY AND BOUNDED FUNCTION

in terms of riemann sum and eplision

limit involing upper and lower sums

norm of refinement is lesser

CAUTION norm of A < NORM B does not imply that a in the refienemnet of B

RIEMANN INTEGRATION THEORY CLOSED BOUNDED INTERVAL AND F IS BOUNDED FUNCTION INITALLY

IMPROPER INTGARAL

Calculus

First fundamental theorem of calculus

Second fundamental theorem of calculus

Improper intrgrak

Type1

Type 2

Infinite. Discontinuity at a point in between or end point of the limits DOUBT only infinite discontinuity?????

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Both type 1 and type 2

Break into two integrals one of each type

multiple integrals

double integrals

triple integrals

function of two variables

APPLICATION

because of that domain is now a area or surface

function of three variables

so Domain is volume in 3D space

important skills

Analyze Domain

plot it

transform it using the change of variables

change the order of integration and its limits

Change of variable OR transformation

use of Jacobian

what is it

gives the appropriate factor

typical

polar

spherical

cylindrical

how to decide which transformation is useful

don't forget the modulus

APPPLICATION

finding the area between two curves

finding the volume of solid

finding the volume of solid

finding

typical

described by rotating a curve along an axis

evaluate Beta Gamma function s

finding gamma half

finding the relation between beta and gamma functions

evaluate some single variable integrals

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typical partitions

i/in

GP a gp like 1 ,r,,r2,,r3,,,,,,,rn

Proving is or is not

Use of inequalities

Disprove sin1/x

Prove xsqaure

Sandwich

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

Limit comparison test

Dirchlet test

Both integrands must be non negative in the required interval

Same conclusions as in series

Only single implicqtion

Both positive function in that domain

Limit find

Zero

Finite

Trick if both negative then test the convergence of their negative cpunterparts

Inifinity

Absolutely convergence

Many times break the integral so as to apply the tests

Example the function may be continous on some part of domain

One function is bounded and montonic on starting to infinity

Other functions integral is bounded

Be careful sometimes it looks infinite discontinuity but we have to check that by finding the limit

If substituting do indefinite don't keep chaniiging the limits

Eplison greater than zero means that limit eplision tends to zero plus

USE IT AS A TEST

IF SOMETHING IS ABSOLUTELY CONVERGENT THEN IT IMPLIES CONVERGENCE

equation of cone 3D

equation cardidoid

equation of cylinder

parabloid

intersection of planes

tetrahedron

cone z = sqrt(x2+y2)

z= x2 +y2

x2 + y2 = constant

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

Usefyl in sin cos where bounded integrals

Be careful to check the discontinuous points in between the interval

Not converge vs diverge

Limit at infinity must be zero

Difficult integration

Change order

Change coordinates

Multiple integrals

Square the integral

Then write in two coordinates same integral

Convert to multiple integral

Area of domain

If fxy is separable into gx and hy and rectangular doamin

use symmetry

xregular y regular