INTERGRAL CALCULUS Flashcards
a branch of mathematics concerned with the theory and applications (as in the determination of lengths, areas, and volumes and in the solution of differential equations) of integrals and integration
integral calculus
result of the integration is called
the integral
two types of integral
definite and indefinite
before integration, the function to be integrated is called
integrand
integral of the ff: sin x = cos x = tan x = cot x = csc x = sec x =
sin x = -cos x + C cos x = sin x + C tan x = -ln |cos x| + C cot x = ln |sin x| + C csc x = -ln |csc x + cot x| + C sec x = ln |sec x + tan x| + C
integral of the ff:
∫ log(a) xdx = ?
∫ ln xdx = ?
∫ log(a) xdx = x ( log(a) x - log(a) e ) + C
∫ ln xdx = x ln(x) -x +C
∫ sec^2 xdx= ∫ csc^2 xdx= ∫ sec x tan xdx= ∫ csc x cot xdx= ∫ 1/(x^2 +1) dx = ∫ sinh xdx = ∫ cosh xdx = ∫ 1/x dx= ∫ b^x dx= ∫ 1/√(1-x^2) dx=
∫ sec^2 xdx= tan x + C ∫ csc^2 xdx= -cot x + C ∫ sec x tan xdx= sec x + C ∫ csc x cot xdx= -csc x + C ∫ 1/(x^2 +1) dx = tan^-1 x + C ∫ sinh xdx = cosh x + C ∫ cosh xdx = sinh x +C ∫ 1/x dx= ln |x| + C ∫ b^x dx= b^x / ln(b) + C ∫ 1/√(1-x^2) dx= sin^-1 x +C
also known as u-substitution or change of variables, is a method for evaluating integrals and antiderivatives.
subtitution rule
given the ff expression, give their respective trigonometric subtitution for integration and the right identity to be applied
√(a^2 - x^2) , subtitution=? identity=?
√(a^2 + x^2), subtitution=? identity=?
√(x^2 - a^2), subtitution=? identity=?
√(a^2 - x^2) ,
subtitution: , let x = a sin θ
identity: ( 1- sin^2 θ) = cos θ
√(a^2 + x^2),
subtitution: let x = atan θ
identity: 1 + tan^2 θ = sec^2 θ
√(x^2 - a^2),
subtitution: let x = asecθ
identity: sec^2 θ -1 = tan^2 θ
long solution…..
solve for
∫ √(9-x^2)/ x^2 dx
ans : - cot θ - θ + C
step 1 : use trigo sub.
√(a^2 - x^2) ,
subtitution: , let x = a sin θ
identity: ( 1- sin^2 θ) = cos θ
step 2: use to simplify
cots^2 θ + 1 = csc^2θ
step 3: use ;
∫ csc^2 xdx= -cot x + C
step4 : answer is
ans : - cot θ - θ + C
integration by parts formula and
how to know the order of choosing u and dv
∫ vdu = uv- ∫ vdu
LIATE
P= f(x,y)
∂P/∂x means that?
derivative of the function of x and y with respect to x,
x is differntiated and y is constant.
how do you solve a miltiple integration.
solve first the inner integration and move outwards:
- take note of the dx , dy ,dz to know which is your variable and constants.
how do you solve the area under the curve.
give the steps
1 visualize the curve
2 choose a strip (vertical or horizontal)
* strips must touch both lines
3 reqrite equation
* horizontal = dy : stacking multiple strips vertically
* vertical strip= dx ; stacking multiple strips
horizontally
4 determine the limits.
5 solve….
A= ∫(x2-x1) [ Yupper - Ylower]dx
or
A= ∫(y2-y1) [ Xright - Xleft]dy
how to solve if there are three curves involved
just to know the intersection and solv efor the area 2 lines at a time
how to solve if there is multiple area bwet two curves.
just use absolute
A= ∫(x3-x1) | f(x) - g(x)| dx
formula for the area of a sector
A = 1/2 r^2 θ
**a sector is used instead of rectangle when integrating with polar coordinate to find area under a curve.
A = ∫(θ2-θ1) [ 1/2 r^2 θ dθ
when evaluating an area under a curve for POLAR COORDINATES, how do you determing the upper limit and the lower limit?
for rectangular, you use upper- lower & righ - left.
but with polar curves, you do clockwise like (330°-210°)
- use radian for ease
- 5
HOW DO YOU SOLVE THE AREA UNDER TWO CRUVES?
A = ∫(θ2-θ1) [ 1/2 ( (r outer)^2 - (r inner)^2 ) dθ
how do you solve the arc length by integration
A = ∫(x2-x1) srt (1 + (dy/dx)^2) dx
A = ∫(y2-y1) srt (1 + (dx/dy)^2) dy
A = ∫(θ2-θ1) srt (r^2 + (dr/dθ)^2) dθ
surface area of a cruve when rotated upon an axis is equal to? as 1st proposed by pappus
SA = (S) (Circumference) SA = S 2πr
S= arc length
A= area
& circumference = 2πr
solving for the surface area stepsss
step 1 identify thrugh what axis or line is the function being rotated to and for the equation SA= S (2πr) and use one of the fomula form depending on that
A = ∫(x2-x1) srt (1 + (dy/dx)^2) dx
A = ∫(y2-y1) srt (1 + (dx/dy)^2) dy
A = ∫(θ2-θ1) srt (r^2 + (dr/dθ)^2) dθ
step 2 solve for dx/dy or dy/dx or dr/dθ
step 3 make sure that the integran is in terms of only one variable, and the limit values are in term of that as well.
step 4: simplify
2nd proposition of pappus. if an area is rotated upon an axis it will generate a volume equal to?
V =( Area) (circumference)
V= A (2πr)
integration formula in finding the volume
WASHER METH0OD.
v = ∫(x2-x1) π(r outer ^2- r inner^2) dx
**ONLY TO BE USE IF THE DIFFERNTIAL STRIP IS PERPENDICULAR TO THE AXIS OF ROTATION
**make sure that the integran is in terms of only one variable, and the limit values are in term of that as well.
integration formula in finding the volume
volume shell method
∫(x2-x1) 2πx (Yupper - Ylower) dx
∫(y2-y1) 2πx (Xright - Xleft) dy
**differntial strip is parallel to the axis of rotation
an expression involving the product of a perpendicular distance and physical quantity about a point, plane or axis
MOMENT
MOMENT FORMULA
parallel:
∫(x2-x1) x (Yupper - Ylower) dx
perpendicular:
∫(x2-x1) 1/2 (Yupper^2 - Ylower^2) dx
parallel:
∫(y2-y1) y (Xright - Xleft) dy
perpendicular:
∫(y2-y1) 1/2 (Xright^2 - Xleft^2) dy
In mathematics and physics, _______is the arithmetic mean position of all the points in the figure. Informally, it is the point at which a cutout of the shape could be perfectly balanced on the tip of a pin.
the centroid or geometric center of a plane figure
FORMULA FOR CENTROID FOR ANY SHAPE
x= My /Area y= Mx/ area
**NEVER SHIFT YOUR DIFFERENTIAL STRIP IN THE MIDDLE OF SOLVING YOUR CENTROID FROM x VARI TO Y var.
Newton’s first law states that if a body is at rest or moving at a constant speed in a straight line, it will remain at rest or keep moving in a straight line at constant speed unless it is acted upon by a force.
this is also called?
law of inertia
is a measure of the force that can cause an object to rotate about an axis.
TORQUE
MOMENT OF INERTIA (I) FORMULA
derive from:
F= (m) (a)
τ = I/α
I = TORQUE (τ) / ANGULAR ACCELERATION α = force (F) radius (r) / acceleration(a)/ radius(r) = (F)(r^2)/ a =
(m)(r^2) = I
I= ∫ r^2dA
dA - this is the area
solving for moment of inertia
I= ∫ r^2dA
- always use parallel with axis
- r= distance to axis of rotation , most likely x or y axis (r=x,r=y)
- dA- (use area formula upper-lower, right-left, depending on the strip of choice)
density
of pure water
vs
saltwater
(ρ)=
purewater = 1000Kg/ m^3 saltwater = 1030Kg/ m^3
liquid pressure formula
P= (ρ)gh P= F/A
formula for hydrostatic force
since P = F/A = (ρ)gh
then
F= (ρ)ghA
h= height
if the center is unknown, the hydrostatic force =?
F= ∫ γ h dA γ= ρg
**choose a parallel strip always
work – fluids
formula when centroid is known….
W=fd d=h f=mg W= mgh ρ=m/v W= ρvgh γ= ρg
W= γvh ————————–
work – fluids
formula when centroid is unknown….
dm= ρdV ghdm= ghρdV W=mgh W = ghρdV γ= ρg W = ∫ γhdV ---------------------------
HOOKES LAW,
PRESSING OR COMPRESSING A SCPRING`
W = ∫(x2-x1) kxdVx k=n/m= the proportionality constant = (rate of force over distance to pul/push the spring, unit weight)
work – lifting a weight. (chain/rope)
work = weight (w) y (height) work = work(load) + work(rope)
work(load) = ∫ W(load) dy work(rope) = ∫ (W(rope) - ωy)dy ω= unit wight [N/m] (weight over length)
WORK= ∫ (W(load) + W(rope) - ωy) dy ————————–
average value of a continuous fucntion
f(ave) = 1/(b-a) ∫(b-a) f(x)dx
f(c) = f(ave)
** f(c) is a function of c not the value of c itself!!
criteria of probability density function (PDF) of a continuous probability
- *** f(x) ≥ 0 (theres is no 0 probability
* ** ∫(∞ - -∞) f(x) dx =1 (area of probability=1)
COMMON PDF’S
UNIFORM DISTRIBUTION
EXPONENTIAL DISTRIB
NORMAL DISTRI
MEAN - EXPECTATION PROBABILITY μ
μ = ∫(∞ - -∞) xf(x) dx ------------------------ μ= aligned with centroid= mean
variance PROBABILITY σ^2
σ^2 = variance σ^2= E((X-μ)^2
μ = E(x)
so;
σ^2= E(X^2) - μ^2———————–
or
σ^2= E(X^2) - E(X)^2 ————
standard ddeviate σ
σ = sqrt(variance)
