Curvilinear Figures and Solids

Circles

For a circle of radius r: diameter = 2r circumference = 2πr area = πr 2

For an arc subtending an angle of measure θ: arc length = —–πr θ

180 For a sector cut off by angle of measure θ:

area = —–πr θ 2 360

For a segment cut off by angle of measure θ: area = —–πr θ 2 ––r 1 2 sinθ

Other Planar Curvilinear Figures

For an annulus with outer radius a and inner radius b: area = π(a 2 –b 2 )

For an ellipse with semi-axes a and b: area = πab

For a cycloid formed from a circle with radius r rolling along a line:

arc length of one arch = 8r area under one arch = 3πr 2

Spheres

For a sphere of radius r: surface area = 4πr 2 volume = – πr 4 3

3 For a spherical cap of height h on a sphere of radius r:

surface area = 2πrh volume = – πh 1 3 (3r – h)

Cylinders

For a right circular cylinder with radius r and height h: lateral surface area = 2πrh volume = πr 2 h

(continues)

CHARTS & TABLES

Curvilinear Figures and Solids Beginning – Ending

Curvilinear Figures and Solids Beginning – Ending CHARTS & TABLES Curvilinear Figures and Solids (continued)

Cones

For a right circular cone with radius r, slant height l, and height h: —— –

lateral surface area = πr √ r 2 + h 2 = πrl

volume = –πr 1 2 h 3

For a frustum of a right circular cone with radii a and b, slant height l, and height h: ——

2 lateral surface are = π(a + b) ––– √ h + (b – a) 2 = π(a + b)l

volume = –πh(a 1 2 + ab + b 2 )

Torus

For a torus with inner radius a and outer radius b:

surface area = π 2 (b 2 –a 2 )

volume = –π 1 2 (a + b)(b – a) 2 3

Ellipsoid

For an ellipsoid with semi-axes a, b, and c: volume = –πabc 4

Beginning – Ending Curvilinear Figures and Solids CHARTS & TABLES

CHARTS & TABLES

Beginning – Ending Conic Sections

Conic Sections

A conic section is the locus of a point P moving so that its distance from a fixed point, the focus, divided by its distance from a fixed line, the directrix, is a constant ε, the eccentricity.

The intersection of a A plane perpendicular

A plane that intersects both cone with . . .

A plane that intersects the cone

A plane parallel to the

nappes of the cone The locus of all

to the axis of the cone

in a simple closed curve

generator of the cone

Distance from a fixed point, Distances from two fixed points whose . . .

Distance from a fixed

Distances from two fixed

point, the center, is a

points, the foci, have a constant

points, the foci, have a

the focus, is equal to its

constant sum

distance from a fixed

constant difference

point, the directrix

Equation center x 2 +y 2 =r 2 x 2 y 2 y =–x 1 2 x 2 y 2 (0, 0)

–+–=1 a 2 b 2 4a –––=1 a 2 b 2 Equation center

0 0 = ——— 4a 0 ——— – ——— = 1 b a 2 b 2 Comments

(x – x 0 ) 2 + (y – y 0 ) 2 =r 2 (x – x 0 ) 2 (y – y 0 ) 2 (x – x 0 ) 2 (x – x 0 ) 2 (y – y 0 ) 2 (x ,y )

——— + ——— = 1 a 2 2 y –y

radius = r

If a > b, the major axis is

If a > 0, the parabola opens The hyperbola opens on the

horizontal. If a < b, the

upward. If a < 0, the

left and the right.

major axis is vertical.

parabola opens

length of horizontal

length of horizontal axis = 2a

downward.

axis = 2a

length of vertical axis = 2b Focus

length of vertical axis = 2b

(x 0 ± √ a 2 +b 2 ,y 0 ) Directrix

0 a 2 √a 2 +b 2 x=x 0 ± ——— √ a 2 +b 2 Eccentricity

a a ε = – = ——— a a Asymptotes

y –y 0 = ±–(x – x b a 0 ) The second order

b equation ax –<0 + by + cxy + dx + ey + f = 0 gives this conic section if:

Tangents and their A tangent is

A tangent makes equal properties

A tangent makes equal

angles with the line from angles with the rays from radius at the point of

perpendicular to a

angles with the rays from

the focus to the point of the point of tangency to tangency.

the point of tangency to the

foci.

tangency and a line

the foci.

perpendicular to the directrix through the point of tangency.

Found in nature Ripples in a pond

Planetary orbits

Trajectory of object thrown Trajectory of a comet that

passes the sun only once Applications

into the air

Wheels

Whispering gallery; medical

Golden Gate Bridge;

Loran; nuclear power

treatment of kidney stones

satellite dishes; solar

cooling towers;

oven

hyperboloidal gears

CHARTS & TABLES

Conic Sections Beginning – Ending

Beginning – Ending Analytic Geometry CHARTS & TABLES Analytic Geometry

Plane Analytic Geometry

midpoint of the segment connecting points (x y

1 ,y 1 ) and (x 2 ,y 2 ): ———– 1 +x 2 1 +y  2  2 , ———– 2  

distance between points (x 1 ,y 1 ) and (x 2 ,y 2 ): √ (x 1 –x 2 ) 2 + (y 1 –y 2 ) 2 y 2 –y 1

slope of the line passing through points (x 1 ,y 1 ) and (x 2 ,y 2 ): ——– x 2 –x 1

slope-intercept equation of the line with slope m and y-intercept (0, b): y = mx + b

two-point equation of the line passing through points (x y ,y ) and (x ,y ): y = ——–(x – x 2 –y 1 1 1 2 2 x 2 –x 1 1 )+y 1

equation of the line passing through the point (x 1 ,y 1 ) with slope m: y = mx + (y 1 – mx 1 ) The lines with slopes m 1 and m 2 are perpendicular if m 1 m 2 = –1.

| Ax

—————– 1 + By 1 distance from point (x +C 1 ,y 1 ) to line Ax + By + C = 0: | 2 | √ A 2 +B |

tangent of the angle ψ between line y = m 1 x +b 1 and y = m 2 x +b 2 : tanψ = ———– m 2 –m 1+m 1 1 m 2

x +x +x y +y +y z +z +z

centroid of the triangle with vertices (x 1 ,y 1 ), (x 2 ,y 2 ), and (x

3 ,y 3 ):  ———–—— , ———–—— , ———–——   3 3 3 

area of the triangle with vertices (x 1 ,y 1 ), (x 2 ,y 2 ),and (x 3 ,y 3 ):

| x 1 y 1 1 |

| 1– det  x 2 y 2 1 | = 1– (x 1 y 2 +x 2 y 3 +x 3 y 1 –x 1 y 3 –x 2 y 1 –x 3 y 2 )

| 2 x 3 y 3  1 | | 2 |

The point (x, y) in rectangular coordinates is the same as point (r, θ) in polar coordinates if : x = r cos θ and y = r sin θ or

r = and x 2 √ 2 +y θ = tan –1 (y/x)

Solid Analytic Geometry

distance between points (x 1 ,y 1 ,z 1 ) and (x 2 ,y 2 ,z 2 ): √ (x 1 –x 2 ) 2 + (y 1 –y 2 ) 2 + (z 1 –z 2 ) 2

equation of the plane passing through points (a, 0, 0), (0, b, 0), and (0, 0, c): – + – + – = 1 x y z a b c equation of the line passing through point (x 1 ,y 1 ,z 1 ) perpendicular to the plane

Ax + By + Cz + D = 0: ——– = ——– = ——– x –x 1 y –y 1 z –z 1 A B C

distance from the point (x ,y ,z ) to the plane Ax +By + Cz + D = 0: | Ax 1 —————–—– + By 1 + Cz 1 1 +D 1 1 | | √ A 2 +B 2 +C 2 |

The point (x, y, z) in rectangular coordinates is the same as point (r, θ, z) in cylindrical coordinates if: x = r cos θ, y = r sin θ, and z = z

r =, x 2 +y √ 2 θ = tan –1 (y/x), and z = z The point (x, y, z) in rectangular coordinates is the same as the point (r, θ, φ) in spherical coordinates if: x = r sinθ cos φ, y = r sin θ cos φ, and z = r cos θ or

–1  ————–— z r  = √ x +y +z , φ = tan (y/x), and φ = cos

 2 √ x +y 2 +z 2 

Analytic Geometry Beginning – Ending CHARTS & TABLES

CHARTS & TABLES

Trigonometry Trigonometry

Curvilinear Figures and Solids