Coordinate Systems

Question 0

Polar axis

Answer 0

X-axis in the coordinate system

Question 1

Pole

Answer 1

Origin of the coordinate system

Question 2

The radial coordinate (r)

Answer 2

The distance from the origin

Question 3

Angular coordinate (θ)

Answer 3

The angle measure from the polar axis to the radial coordinate

Question 4

Calculate ‘x’ from ‘r’ and θ

Answer 4

x=r*cos(θ)

Question 5

Calculate ‘y’ from ‘r’ and θ

Answer 5

y=r*sin(θ)

Question 6

Calculate ‘r’ from ‘x’ and θ

Answer 6

r=x/(cos θ)

Question 7

Calculate ‘r’ from ‘y’ and θ

Answer 7

r=y/(sin θ)

Question 8

Calculate ‘r’ from ‘x’ and ‘y’

Answer 8

r=√(x^2+y^2)

Question 9

Calculate ‘x’ from ‘r’ and ‘y’

Answer 9

x=√(r^2-y^2)

Question 10

Calculate ‘y’ from ‘x’ and ‘r’

Answer 10

y=√(r^2-x^2)

Question 11

Calculate θ from ‘y’ and ‘x’

Answer 11

θ=tan^(-1)(y/x)

Question 12

Calculate θ from ‘y’ and ‘r’

Answer 12

θ=sin^-1(y/r)

Question 13

Calculate θ from ‘x’ and ‘r’

Answer 13

θ=cos^-1(x/r)

Question 14

Calculate ‘x’ from ‘y’ and θ

Answer 14

x=y/tanθ

Question 15

Calculate ‘y’ from ‘x’ and ‘θ’

Answer 15

y=x*tanθ

Question 16

Cardioid

Answer 16

Graph of the radius with a change in θ

Question 17

Lemniscate

Answer 17

Graph of the (radius)^2 with a change in θ

Question 18

Cartesian Coordinates in a Plane

Answer 18

Describe the location of a point in terms of (x,y)

Question 19

Polar Coordinates

Answer 19

Describe the location of a point in terms of (r,θ)

Question 20

Slope of a tangent line in polar coordinates

Answer 20

mTan=[f’(θ)sinθ+f(θ)cosθ]/[f’(θ)cos-f(θ)sinθ]

Question 21

Using polar coordinates to approximate integrals

Answer 21

∫1/2*(f(θ)^2-g(θ)^2)dθ
On the interval from [θi,θf]
See pg 660

Question 22

Cartesian coordinates for horizontal ellipses

Answer 22

1=(x^2/a^2)+(y^2/b^2)

b- √(a^2-c^2)
a- vertices at (0,±a) or (±a,0)
c- foci at (0,±c) or (±c,0)

Question 23

Cartesian coordinates for vertical ellipses

Answer 23

1=(y^2/a^2)+(x^2/b^2)

b- √(a^2-c^2)
a- vertices at (0,±a) or (±a,0)
c- foci at (0,±c) or (±c,0)

Question 24

Cartesian coordinates for horizontal hyperbolas

Answer 24

1=(x^2/a^2)-(y^2/b^2)

b- √(c^2-a^2)
a- vertices at (0,±a) or (±a,0)
c- foci at (0,±c) or (±c,0)

Question 25

Cartesian coordinates for vertical hyperbolas

Answer 25

1=(y^2/a^2)-(x^2/b^2)

b- √(c^2-a^2)
a- vertices at (0,±a) or (±a,0)
c- foci at (0,±c) or (±c,0)

Question 26

Polar equation for parabolic curves of d>(x=0)

Answer 26

r=εd/(1+εcosθ)

Question 27

Polar equation for parabolic curves of d<(x=0)

Answer 27

r=εd/(1-εcosθ)

Question 28

Polar equation for parabolic curves of d>(y=0)

Answer 28

r=εd/(1+εsinθ)

Question 29

Polar equation for parabolic curves of d<(y=0)

Answer 29

r=εd/(1-εsinθ)

Question 30

Cartesian coordinates for parabola

Answer 30

y=x^2/(4r)

Question 31

Arc Length for polar curves

Answer 31

L=∫√[f(θ)^2+f’(θ)^2]dθ

Question 32

xy-Coordinates for the center of mass

Answer 32

X-coordinate from zero position=(∫∫xp(x,y)/(∫∫p(x,y)
Y-coordinate from zero position=(∫∫yp(x,y)/(∫∫p(x,y)

Density functions, and the product of their linear position

Question 33

Center of mass in a 3d coordinate plane

Answer 33

X-coordinate from zero position=(∫∫∫xp(x,y,z)/(∫∫∫p(x,y,z)
Y-coordinate from zero position=(∫∫∫yp(x,y,z)/(∫∫∫p(x,y,z)
Z-coordinate from zero position=(∫∫∫z*p(x,y,z)/(∫∫∫p(x,y,z)

Density functions, and the product of their linear position