Section 7.7:
Approximate Integration
Midpoint rule:
euler(f(x*i)deltax)
integ(f(x)dx~=M_n)from(a->b)
where deltax=(b-a)/n
n=number of rectangles

Trapezoidal Rule:
integ(f(x)dx~=T_n)from(a->b)
where deltax=(b-a)/n
and xi=a+ideltax

Simpsons rule:
Uses parabolas
integ(f(x)dx)from(a->b)
where
(h/3)[f(a)+4f((a+b)/2)+f(b)]

Error:
E_T=integ(f(x)dx)from(a->b) - T_n
# trapezoidal rule^

E_M=integ(f(x)dx)from(a->b) - M_n

Basically subtract the approx from the actual value

Section 8.1:
Arclength
L=lim(n->inf) euler(i=1,n,|P_i-1P_i)
P_i-1P_i=
sqrt((x_i-x_(i-1))^2 + (y_i-y_(i-1))^2)
Arc length formula:
L=integ(sqrt(1+(f'(x))^2))dx from (a->b)

Section 8.2:
Area of a Surface of Revolution
L=integ(ds)
C=2pir
so 2pi*f(x) will be for rotating around x axis.

ds=sqrt(1+(dy/dx)^2)
or
ds=sqrt(1+(dx/dy)^2)

for rotating around x axis
general form:
S=integ(2pi*y)ds from (a->b)
so two options:
S=integ(2pi*y*sqrt(1+(dy/dx)^2))dx from (a->b)
or
S=integ(2pi*y*sqrt(1+(dx/dy)^2))dy from(c->d)

for rotating around y axis
general form:
S=integ(2pi*x)ds from (a->b)
radius is x, or g(y)
so two options:
S=integ(2pi*x*sqrt(1+(dy/dx)^2))dx
S=integ(2pi*x*sqrt(1+(dx/dy)^2))dy

Section 8.3:
Applications of Physics and Engineering
Finding the centroid

m_1(avg(x)-x_1)=m_2(x_2-avg(x))
mass=density*area = P*A
A=integ(f(x)dx)from(a->b)

Section 10.1:
Curves defined by parametric functions
You know this.

Section 10.2:
Calculus with parametric curves
Chain rule:
(dy/dt)=(dy/dx)*(dx/dt)
If (dx/dt)=/=0 we can find (dy/dx)

(dy/dx)=(dy/dt)/(dx/dt)
IMPORTANT:
(d/dx)(y)=((d/dt)(y))/((d/dt)(x))

Also, ((d^2)y)/(d(x^2)) which can be found by replacing y.
((d^2)y)/(d(x^2))=(d/dx)(dy/dx)
=((d/dt)(dy/dx))/((d/dt)(x))
=((d/dt)((dy/dt)/(dx/dt)))/(dx/dt)

Arc length of a parametric curve is:
if F' is continuous
then
L=integ(sqrt(1+(f'(x))^2))dx from (a->b)

Section 10.3:
Areas and lengths in polar coordinates
A=(1/2)r^2 theta
A=integ((1/2)[f(theta_i *)]^2 deltatheta)

example:
r=3sintheta
r=2-sintheta
A=integ((1/2))

Section 10.4:


Section 10.5:
Conic sections:
Parabola with focus (0,p)
directorex = y=-p
X^2 = 4 p Y (opens up(+p)/down(-p))
Y^2 = 4 p X (opens left(-p)/right(+p))

Equation of ellipse:
(x^2/a^2)+(y^2/b^2)=1
a>=b>0
verts= +-a,0
foci= +-c,0
c^2 = a^2 - b^2

Equation of hyperbola:
(x^2/a^2)-(y^2/b^2)=1 (no y-ints)
vertex:(+-a,0)
foci:(+-c,0)
c^2 = a^2 + b^2
or
(y^2/a^2)-(x^2/b^2)=1 (no x-ints)
vertex:(0,+-a)
foci:(0,+-c)
c^2 = a^2 + b^2

Section 10.6:
Conic sections in polar coordinates
Let F be a fixed point (called the focus)
and let l be a fixed line (called the directrix) in a plane
Let e be a fixed positive number (called the eccentricity)
The set of all points P in the plane such that
(|PF|) # Distance between points in plane P and F
/
(|Pl|) # Distance between points in plane P and a fixed line
=
e # eccentricity

Circle: e=0
Ellipse: 0<e<1
Parabola: e=1
Hyperbola: e>1

Polar equation of the form:
r=(e*d)/(1+-e*cos(theta))
or
r=(e*d)/(1+-e*sin(theta))

if +cos(theta) then directorex is x=d
if -cos(theta) then directorex is x=-d

if +sin(theta) then directorex is y=d
if -sin(theta) then directorex is y=-d