Harold’s AP Calculus BC
Rectangular – Polar – Parametric
“Cheat Sheet”
4 March 2015
Point
Rectangular
Polar
Parametric
𝑓(𝑥) = 𝑦
(𝑥, 𝑦)
(𝑎, 𝑏)
(𝑟 , 𝜃) or 𝑟 ∠ 𝜃
𝑥 = 𝑟 𝑐𝑜𝑠 𝜃
𝑟2 = 𝑥2 + 𝑦2
𝑦 = 𝑟 𝑠𝑖𝑛 𝜃
𝑟 = ± √𝑥 2 + 𝑦 2
Point (a,b) in Rectangular:
𝑥(𝑡) = 𝑎
𝑦(𝑡) = 𝑏
𝑡 = 3𝑟𝑑 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒, 𝑢𝑠𝑢𝑎𝑙𝑙𝑦 𝑡𝑖𝑚𝑒
< 𝑎, 𝑏 >
•
𝑦
𝑡𝑎𝑛 𝜃 = (𝑥 )
𝑦
𝜃 = 𝑡𝑎𝑛−1 (𝑥 )
Slope-Intercept Form:
𝑦 = 𝑚𝑥 + 𝑏
< 𝑥, 𝑦 > = < 𝑥0 , 𝑦0 > + 𝑡 < 𝑎, 𝑏 >
< 𝑥, 𝑦 > = < 𝑥0 + 𝑎𝑡, 𝑦0 + 𝑏𝑡 >
where
< 𝑎, 𝑏 > = < 𝑥2 − 𝑥1 , 𝑦2 − 𝑦1 >
Point-Slope Form:
𝑦 − 𝑦0 = 𝑚 (𝑥 − 𝑥0 )
Line
General Form:
𝐴𝑥 + 𝐵𝑦 + 𝐶 = 0
𝑤ℎ𝑒𝑟𝑒 𝐴 𝑎𝑛𝑑 𝐵 ≠ 0
𝑥(𝑡) = 𝑥0 + 𝑡𝑎
𝑦(𝑡) = 𝑦0 + 𝑡𝑏
𝑓(𝑥) = 𝑓 ′ (𝑎) 𝑥 + 𝑓(0)
where 𝑚 = 𝑓’(𝑎)
Copyright © 2011-2014 by Harold Toomey, WyzAnt Tutor
𝑚=
1
∆𝑦
𝑦2 − 𝑦1 𝑏
=
=
∆𝑥 𝑥2 − 𝑥1 𝑎
Rectangular
Polar
Parametric
𝑛𝑥 (𝑥 − 𝑥0 )
+ 𝑛𝑦 (𝑦 − 𝑦0 )
+ 𝑛𝑧 (𝑧 − 𝑧0 ) = 0
𝒓 = 𝒓0 + 𝑠𝒗 + 𝑡𝒘
where:
 s and t range over all real
numbers
 v and w are given vectors
defining the plane
 𝒓𝟎 is the vector representing the
position of an arbitrary (but
fixed) point on the plane
Vector form:
𝒏 ⦁ (𝒓 − 𝒓0 ) = 0
Plane
r = a (constant)
𝜃 = 𝜃 [0, 2𝜋] 𝑜𝑟 [0, 360°]
𝑥(𝑡) = 𝑟 𝑐𝑜𝑠(𝑡) + ℎ
𝑦(𝑡) = 𝑟 𝑠𝑖𝑛(𝑡) + 𝑘
[𝑡𝑚𝑖𝑛 , 𝑡𝑚𝑎𝑥 ] = [0, 2𝜋)
𝑥2 + 𝑦2 = 𝑟2
(𝑥 − ℎ)2 + (𝑦 − 𝑘)2 = 𝑟 2
Circle
General Form:
𝐴𝑥 2 + 𝐵𝑥𝑦 + 𝐶𝑦 2 + 𝐷𝑥 + 𝐸𝑦 + 𝐹
=0
𝑤ℎ𝑒𝑟𝑒 𝐴 = 𝐶 𝑎𝑛𝑑 𝐵 = 0
(ℎ, 𝑘) = 𝑐𝑒𝑛𝑡𝑒𝑟 𝑜𝑓 𝑐𝑖𝑟𝑐𝑙𝑒
(ℎ, 𝑘)
Focus and Center:
(ℎ, 𝑘)
Copyright © 2011-2014 by Harold A. Toomey
.
2
Rectangular
2
Polar
Parametric
.
2
(𝑥 − ℎ)
(𝑦 − 𝑘)
+
=1
2
𝑎
𝑏2
General Form:
𝐴𝑥 2 + 𝐵𝑥𝑦 + 𝐶𝑦 2 + 𝐷𝑥 + 𝐸𝑦 + 𝐹
=0
where 𝐵2 − 4𝐴𝐶 < 0 𝑜𝑟 𝐴𝐶 > 0
Center: (ℎ, 𝑘)
Vertices:
(ℎ ± 𝑎, 𝑘)
(ℎ, 𝑘 ± 𝑏)
Foci: (ℎ ± 𝑐, 𝑘)
Ellipse
Focus length, c, from center:
𝑐 = √𝑎2 − 𝑏 2
𝑟(𝜃) =
𝑥(𝑡) = 𝑎 𝑐𝑜𝑠(𝑡) + ℎ
𝑦(𝑡) = 𝑏 𝑠𝑖𝑛(𝑡) + 𝑘
[𝑡𝑚𝑖𝑛 , 𝑡𝑚𝑎𝑥 ] = [0, 2𝜋]
𝑎𝑏
(ℎ, 𝑘) = 𝑐𝑒𝑛𝑡𝑒𝑟 𝑜𝑓 𝑒𝑙𝑙𝑖𝑝𝑠𝑒
√(𝑏 𝑐𝑜𝑠 𝜃)2 + (𝑎 𝑠𝑖𝑛 𝜃)2
relative to center
---------------------------------------------------Rotated Ellipse:
𝑥(𝑡) = 𝑎 𝑐𝑜𝑠 𝑡 𝑐𝑜𝑠 𝜃 − 𝑏 𝑠𝑖𝑛 𝑡 𝑠𝑖𝑛 𝜃 + ℎ
𝑦(𝑡) = 𝑎 𝑐𝑜𝑠 𝑡 𝑠𝑖𝑛 𝜃 + 𝑏 𝑠𝑖𝑛 𝑡 𝑐𝑜𝑠 𝜃 + 𝑘
𝜃 = the angle between the x-axis and the
major axis of the ellipse
Eccentricity:
𝑐
√𝑎2 − 𝑏 2
𝑒= =
𝑎
𝑎
If B ≠ 0, then rotate coordinate
system:
𝐴−𝐶
𝑐𝑜𝑡 2𝜃 =
𝐵
𝑥 = 𝑥′ 𝑐𝑜𝑠 𝜃 − 𝑦′ 𝑠𝑖𝑛 𝜃
𝑦 = 𝑦′ 𝑐𝑜𝑠 𝜃 + 𝑥′ 𝑠𝑖𝑛 𝜃
New = (x’, y’), Old = (x, y)
rotates through angle 𝜃 from x-axis
Copyright © 2011-2014 by Harold A. Toomey
3
Rectangular
2
Polar
Parametric
2
(𝑥 − ℎ)
(𝑦 − 𝑘)
−
=1
2
𝑎
𝑏2
General Form:
𝐴𝑥 2 + 𝐵𝑥𝑦 + 𝐶𝑦 2 + 𝐷𝑥 + 𝐸𝑦 + 𝐹
=0
where 𝐵2 − 4𝐴𝐶 > 0 𝑜𝑟 𝐴𝐶 < 0
If 𝐴 + 𝐶 = 0, square hyperbola
Center: (ℎ, 𝑘)
Vertices: (ℎ ± 𝑎, 𝑘)
Foci: (ℎ ± 𝑐, 𝑘)
Hyperbola
𝑟=
𝑎(𝑒 2 − 1)
1 + 𝑒 𝑐𝑜𝑠 𝜃
Left-Right Opening Hyperbola:
𝑥(𝑡) = 𝑎 𝑠𝑒𝑐( 𝑡) + ℎ
𝑦(𝑡) = 𝑏 𝑡𝑎𝑛( 𝑡) + 𝑘
[𝑡𝑚𝑖𝑛 , 𝑡𝑚𝑎𝑥 ] = [−𝑐, 𝑐]
(h, k) = vertex of hyperbola
1
1
−𝑐𝑜𝑠 −1 (− ) < 𝜃 < 𝑐𝑜𝑠 −1 (− )
𝑒
𝑒
2
2
𝑐
√𝑎 + 𝑏
𝑤ℎ𝑒𝑟𝑒 𝑒 = =
= sec 𝜃
𝑎
𝑎
Up-Down Opening Hyperbola:
𝑥(𝑡) = 𝑎 𝑡𝑎𝑛(𝑡) + ℎ
𝑦(𝑡) = 𝑏 𝑠𝑒𝑐(𝑡) + 𝑘
[𝑡𝑚𝑖𝑛 , 𝑡𝑚𝑎𝑥 ] = [−𝑐, 𝑐]
(h, k) = vertex of hyperbola
Focus length, c, from center:
𝑐 = √𝑎2 + 𝑏 2
Eccentricity:
𝑐
√𝑎2 + 𝑏 2
𝑒= =
= sec 𝜃
𝑎
𝑎
General Form:
𝑥(𝑡) = 𝐴𝑡 2 + 𝐵𝑡 + 𝐶
𝑦(𝑡) = 𝐷𝑡 2 + 𝐸𝑡 + 𝐹
where A and D have different signs
If B ≠ 0, then rotate coordinate
system:
𝐴−𝐶
𝑐𝑜𝑡 2𝜃 =
𝐵
𝑥 = 𝑥′ 𝑐𝑜𝑠 𝜃 − 𝑦′ 𝑠𝑖𝑛 𝜃
𝑦 = 𝑦′ 𝑐𝑜𝑠 𝜃 + 𝑥′ 𝑠𝑖𝑛 𝜃
New = (x’, y’), Old = (x, y)
rotates through angle 𝜃 from x-axis
Copyright © 2011-2014 by Harold A. Toomey
4
.
Rectangular
Vertical Axis of Symmetry:
𝑥 2 = 4 𝑝𝑦
(𝑥 − ℎ)2 = 4𝑝(𝑦 − 𝑘)
Vertex: (ℎ, 𝑘)
Focus: (ℎ, 𝑘 + 𝑝)
Directrix: 𝑦 = 𝑘 − 𝑝
Horizontal Axis of Symmetry:
𝑦 2 = 4 𝑝𝑥
(𝑦 − 𝑘)2 = 4𝑝(𝑥 − ℎ)
Vertex: (ℎ, 𝑘)
Focus: (ℎ + 𝑝, 𝑘)
Directrix: 𝑥 = ℎ − 𝑝
Parabola
Polar
Parametric
Vertical Axis of Symmetry:
𝑥(𝑡) = 2𝑝𝑡 + ℎ
𝑦(𝑡) = 𝑝𝑡 2 + 𝑘 (opens upwards) or
𝑦(𝑡) = −𝑝𝑡 2 − 𝑘 (opens downwards)
[𝑡𝑚𝑖𝑛 , 𝑡𝑚𝑎𝑥 ] = [−𝑐, 𝑐]
(h, k) = vertex of parabola
Example: 𝑦 = 𝑥 2
𝑟 𝑠𝑖𝑛 𝜃 = 𝑟 2 𝑐𝑜𝑠 2 𝜃
Horizontal Axis of Symmetry:
𝑦(𝑡) = 2𝑝𝑡 + 𝑘
𝑥(𝑡) = 𝑝𝑡 2 + ℎ (opens to the right) or
𝑥(𝑡) = −𝑝𝑡 2 − ℎ (opens to the left)
[𝑡𝑚𝑖𝑛 , 𝑡𝑚𝑎𝑥 ] = [−𝑐, 𝑐]
(h, k) = vertex of parabola
𝑠𝑖𝑛 𝜃
𝑟=
= 𝑡𝑎𝑛 𝜃 𝑠𝑒𝑐 𝜃
𝑐𝑜𝑠 2 𝜃
General Form:
𝐴𝑥 2 + 𝐵𝑥𝑦 + 𝐶𝑦 2 + 𝐷𝑥 + 𝐸𝑦 + 𝐹
=0
where 𝐵2 − 4𝐴𝐶 = 0
or 𝐴𝐶 = 0
Projectile Motion:
𝑥(𝑡) = 𝑥0 + 𝑣𝑥 𝑡
𝑦(𝑡) = 𝑦0 + 𝑣𝑦 𝑡 − 16𝑡 2 feet
𝑦(𝑡) = 𝑦0 + 𝑣𝑦 𝑡 − 4.9𝑡 2 meters
𝑣𝑥 = 𝑣 𝑐𝑜𝑠 𝜃
𝑣𝑦 = 𝑣 𝑠𝑖𝑛 𝜃
If B ≠ 0, then rotate coordinate
system:
𝐴−𝐶
𝑐𝑜𝑡 2𝜃 =
𝐵
𝑥 = 𝑥′ 𝑐𝑜𝑠 𝜃 − 𝑦′ 𝑠𝑖𝑛 𝜃
𝑦 = 𝑦′ 𝑐𝑜𝑠 𝜃 + 𝑥′ 𝑠𝑖𝑛 𝜃
General Form:
𝑥 = 𝐴𝑡 2 + 𝐵𝑡 + 𝐶
𝑦 = 𝐷𝑡 2 + 𝐸𝑡 + 𝐹
where A and D have the same sign
New = (x’, y’), Old = (x, y)
rotates through angle 𝜃 from x-axis
Copyright © 2011-2014 by Harold A. Toomey
5
.
Rectangular
𝑓(𝑥 + ℎ) − 𝑓(𝑥)
ℎ→0
ℎ
𝑓(𝑥)
−
𝑓(𝑐)
𝑓 ′ (𝑐) = lim
𝑥→𝑐
𝑥−𝑐
𝑑𝑦
′ (𝑥)
𝑓
=
= 𝑦′ = 𝐷𝑥
𝑑𝑥
𝑓 ′ (𝑥) = lim
1st
Derivative
2nd
Derivative
𝑑 𝑑𝑦
𝑑2 𝑦
𝑓 ′′ (𝑥) =
( )=
= 𝑦′′
𝑑𝑥 𝑑𝑥
𝑑𝑥 2
Polar
Parametric
𝑑𝑦 𝑑𝑟
𝑑𝑦 𝑑𝜃 𝑑𝜃 𝑠𝑖𝑛 𝜃 + 𝑟 𝑐𝑜𝑠 𝜃
=
=
𝑑𝑟
𝑑𝑥 𝑑𝑥
𝑐𝑜𝑠 𝜃 − 𝑟 𝑠𝑖𝑛 𝜃
𝑑𝜃
𝑑𝜃
𝑑𝑦
𝑑𝑦
= 𝑑𝑡 ,
𝑑𝑥 𝑑𝑥
𝑑𝑡
Hint: Use Product Rule for
𝑦 = 𝑟 𝑠𝑖𝑛 𝜃
𝑥 = 𝑟 𝑐𝑜𝑠 𝜃
𝑑 𝑑𝑦
( )
𝑑2 𝑦
𝑑 𝑑𝑦
𝑑𝜃 𝑑𝑥
=
(
)
=
𝑑𝑥
𝑑𝑥 2 𝑑𝑥 𝑑𝑥
𝑑𝜃
𝑝𝑟𝑜𝑣𝑖𝑑𝑒𝑑
.
𝑑𝑥
≠0
𝑑𝑡
𝑑 𝑑𝑦
( )
𝑑2 𝑦
𝑑 𝑑𝑦
𝑑𝑡 𝑑𝑥
=
(
)
=
𝑑𝑥
𝑑𝑥 2 𝑑𝑥 𝑑𝑥
𝑑𝑡
Riemann Sum:
𝑛
𝑆=∑
𝑖−1
𝑏
Integral
𝐹(𝑥) = ∫ 𝑓(𝑥) 𝑑𝑥 = 𝐹(𝑏) − 𝐹(𝑎)
𝑎
𝑓(𝑦𝑖 )(𝑥𝑖 − 𝑥𝑖−1 )
Left Sum:
1
1
2
𝑆 = ( ) [𝑓(𝑎) + 𝑓 (𝑎 + ) + 𝑓 (𝑎 + )
𝑛
𝑛
𝑛
1
+ ⋯ + 𝑓(𝑏 − )]
𝑛
Middle Sum:
1
1
3
𝑆 = ( ) [𝑓 (𝑎 + ) + 𝑓 (𝑎 + ) + ⋯
𝑛
2𝑛
2𝑛
1
+ 𝑓(𝑏 − )]
2𝑛
Right Sum:
1
1
2
𝑆 = ( ) [𝑓 (𝑎 + ) + 𝑓 (𝑎 + ) + ⋯
𝑛
𝑛
𝑛
+ 𝑓(𝑏)]
Copyright © 2011-2014 by Harold A. Toomey
6
Rectangular
𝑓(𝑓 −1 (𝑥)) = 𝑓 −1 (𝑓(𝑥)) = 𝑥
Inverse
Functions
Inverse Function Theorem:
1
𝑓 −1 (𝑏) = ′
𝑓 (𝑎)
where 𝑏 = 𝑓 ′ (𝑎)
Parametric
−1
if 𝑦 = 𝑠𝑖𝑛 𝜃
𝑖𝑓 𝑦 = 𝑐𝑜𝑠 𝜃
𝑖𝑓 𝑦 = 𝑡𝑎𝑛 𝜃
𝑡ℎ𝑒𝑛 𝜃 = 𝑠𝑖𝑛 𝑦
𝑡ℎ𝑒𝑛 𝜃 = 𝑐𝑜𝑠 −1 𝑦
𝑡ℎ𝑒𝑛 𝜃 = 𝑡𝑎𝑛−1 𝑦
𝜃 = arcsin 𝑦
𝜃 = arccos 𝑦
𝜃 = 𝑎𝑟𝑐𝑡𝑎𝑛 𝑦
if 𝑦 = 𝑐𝑠𝑐 𝜃
if 𝑦 = 𝑠𝑒𝑐 𝜃
𝑖𝑓 𝑦 = 𝑐𝑜𝑡 𝜃
𝑡ℎ𝑒𝑛 𝜃 = 𝑐𝑠𝑐 −1 𝑦
𝑡ℎ𝑒𝑛 𝜃 = 𝑠𝑒𝑐 −1 𝑦
𝑡ℎ𝑒𝑛 𝜃 = 𝑐𝑜𝑡 −1 𝑦
𝜃 = 𝑎𝑟𝑐𝑐𝑠𝑐 𝑦
𝜃 = 𝑎𝑟𝑐𝑠𝑒𝑐 𝑦
𝜃 = 𝑎𝑟𝑐𝑐𝑜𝑡 𝑦
.
𝐿 = ∫ 𝑑𝑠
𝑏
𝐿 = ∫ √1 +
[𝑓 ′ (𝑥)]2
𝑑𝑥
𝑎
Proof :
∆𝑠 = √(𝑥 − 𝑥0 )2 + (𝑦 − 𝑦0 )2
∆𝑠 = √(∆𝑥)2 + (∆𝑦)2
𝑑𝑠 = √𝑑𝑥 2 + 𝑑𝑦 2
Arc Length
Polar
𝑑𝑟 2
2
√
𝑑𝑠 = 𝑟 + ( ) 𝑑𝜃
𝑑𝜃
Circle:
𝐿 = 𝑠 = 𝑟𝜃
𝑑𝑥 2
𝑑𝑠 = √𝑑𝑥 2 + 𝑑𝑦 2 ( 2 )
𝑑𝑥
Proof:
𝐿 = (𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑐𝑖𝑟𝑐𝑢𝑚𝑓𝑒𝑟𝑒𝑛𝑐𝑒) ∙
𝜋 ∙ (𝑑𝑖𝑎𝑚𝑒𝑡𝑒𝑟)
𝑑𝑦 2 2
2
√
𝑑𝑠 = 𝑑𝑥 + ( ) 𝑑𝑥
𝑑𝑥
𝜃
𝐿 = ( ) 𝜋 (2𝑟) = 𝑟𝜃
2𝜋
𝛽
𝐿 = ∫ √(
𝛼
𝛽
𝑑𝑥 2
𝑑𝑦 2
𝑑𝑧 2
√
𝐿 = ∫ ( ) + ( ) + ( ) 𝑑𝑡
𝑑𝑡
𝑑𝑡
𝑑𝑡
𝛼
𝑑𝑦 2
𝑑𝑠 = √𝑑𝑥 2 (1 + ( ) )
𝑑𝑥
𝑑𝑠 = √1 + (
𝑑𝑥 2
𝑑𝑦 2
) + ( ) 𝑑𝑡
𝑑𝑡
𝑑𝑡
𝑑𝑦 2
) 𝑑𝑥
𝑑𝑥
𝐿 = ∫ 𝑑𝑠
Perimeter
Square: P = 4s
Rectangle: P = 2l + 2w
Triangle : P = a + b + c
Copyright © 2011-2014 by Harold A. Toomey
Circle: C = πd = 2πr
Ellipse: 𝐶 ≈ 𝜋(𝑎 + 𝑏)
𝜋
𝑐 2
Ellipse: 𝐶 = 4𝑎 ∫02 √1 − (𝑎) 𝑠𝑖𝑛2 𝜃 𝑑𝜃
7
Rectangular
Square: A = s²
Rectangle: A = lw
Rhombus: A = ½ ab
Parallelogram: A = bh
(𝑏 +𝑏 )
Trapezoid: 𝐴 = 1 2 2 ℎ
Kite: 𝐴 =
𝑑1 𝑑 2
2
Triangle: A = ½ bh
Triangle: A = ½ ab sin(C)
Area
Triangle: 𝐴 =
√𝑠(𝑠 − 𝑎)(𝑠 − 𝑏)(𝑠 − 𝑐),
𝑎+𝑏+𝑐
𝑤ℎ𝑒𝑟𝑒 𝑠 =
Polar
Parametric
𝛽
𝛽
𝐴 = ∫ 𝑔(𝑡) 𝑓 ′ (𝑡) 𝑑𝑡
1
𝐴 = ∫ [𝑓(𝜃)]2 𝑑𝜃
2
𝛼
where 𝑓(𝑡) = 𝑥 and 𝑔(𝑡) = 𝑦
or
x(t) = f(t) and y(t) = g(t)
𝛼
where 𝑟 = 𝑓(𝜃)
Proof:
Area of a sector:
Simplified :
1
𝐴 = ∫ 𝑠 𝑑𝑟 = ∫ 𝑟 ∆𝜃 𝑑𝑟 = 𝑟 2 ∆𝜃
2
where arc length 𝑠 = 𝑟 ∆𝜃
𝛽
𝐴 = ∫ 𝑦(𝑡)
𝛼
Proof:
𝑏
2
Equilateral Triangle: 𝐴 = ¼√3𝑠 2
∫ 𝑓(𝑥) 𝑑𝑥
𝑎
1 𝑏 +𝑏
y = f(x) = g(t)
Frustum: 𝐴 = ( 1 2 ) ℎ
3
2
Circle: A = πr²
Circular Sector: A = ½ r²𝜃
Ellipse: A = πab
Lateral
Surface
Area
Cylinder: S = 2πrh
Cone: S = πrl
𝑏
𝑑𝑥(𝑡)
𝑑𝑡
𝑑𝑡
For rotation about the x-axis:
𝑑𝑓(𝑡)
𝑑𝑡 = 𝑓’(𝑡) 𝑑𝑡
𝑑𝑡
For rotation about the x-axis:
𝑆 = ∫ 2𝜋𝑦 𝑑𝑠
𝑆 = ∫ 2𝜋𝑦 𝑑𝑠
For rotation about the y-axis:
For rotation about the y-axis:
𝑆 = ∫ 2𝜋𝑥 𝑑𝑠
𝑆 = ∫ 2𝜋𝑥 𝑑𝑠
𝑑𝑟 2
2
√
𝑑𝑠 = 𝑟 + ( ) 𝑑𝜃
𝑑𝜃
𝑟 = 𝑓(𝜃),
𝛼≤ 𝜃≤𝛽
𝑑𝑥 2
𝑑𝑦 2
√
𝑑𝑠 = ( ) + ( ) 𝑑𝑡
𝑑𝑡
𝑑𝑡
𝑖𝑓 𝑥 = 𝑓(𝑡), 𝑦 = 𝑔(𝑡), 𝛼 ≤ 𝑡 ≤ 𝛽
𝑑𝑥 =
𝑆 = 2𝜋 ∫ 𝑓(𝑥) √1 + [𝑓 ′ (𝑥)]2 𝑑𝑥
𝑎
Copyright © 2011-2014 by Harold A. Toomey
8
.
Rectangular
Total
Surface
Area
Cube: S = 6s²
Rectangular Box: S = 2lw + 2wh +
2hl
Regular Tetrahedron: S = 2bh
Cylinder: S = 2πr (r + h)
Cone: S = πr² + πrl = πr (r + l)
For revolution about the x-axis:
𝑏
Surface of
Revolution
𝑑𝑦 2
𝐴 = 2𝜋 ∫ 𝑓(𝑥) √1 + ( ) 𝑑𝑥
𝑑𝑥
𝑎
For revolution about the y-axis:
𝑏
𝑑𝑥 2
𝐴 = 2𝜋 ∫ 𝑥 √1 + ( ) 𝑑𝑦
𝑑𝑦
𝑎
Volume
Cube: V = s³
Rectangular Prism: V = lwh
Cylinder: V = πr²h
Triangular Prism: V= Bh
Tetrahedron: V= ⅓ bh
Pyramid: V = ⅓ Bh
Cone: V = ⅓ bh = ⅓ πr²h
Copyright © 2011-2014 by Harold A. Toomey
Polar
Parametric
Sphere: S = 4πr²
Ellipsoid: S = (too complex)
For revolution about the x-axis:
𝛽
𝐴 = 2𝜋 ∫ 𝑟 𝑐𝑜𝑠 𝜃 √𝑟 2 + (
𝛼
For revolution about the x-axis:
𝑏
𝑑𝑟 2
) 𝑑𝜃
𝑑𝜃
𝑑𝑥 2
𝑑𝑦 2
𝐴 = 2𝜋 ∫ 𝑦(𝑡) √( ) + ( ) 𝑑𝑡
𝑑𝑡
𝑑𝑡
𝑎
For revolution about the y-axis:
For revolution about the y-axis:
𝛽
𝑏
𝑑𝑟 2
𝐴 = 2𝜋 ∫ 𝑟 𝑠𝑖𝑛 𝜃 √𝑟 2 + ( ) 𝑑𝜃
𝑑𝜃
𝑑𝑥 2
𝑑𝑦 2
𝐴 = 2𝜋 ∫ 𝑥(𝑡) √( ) + ( ) 𝑑𝑡
𝑑𝑡
𝑑𝑡
𝛼
4
3
4
=3
𝑎
Sphere: 𝑉 = 𝜋𝑟 3
Ellipsoid: V
πabc
9
.
Rectangular
Polar
Parametric
Disc Method - Rotation about the xaxis:
𝑏
𝑉 = ∫ 𝜋 [𝑓(𝑥)]2 𝑑𝑥
𝑎
𝑏
Cylindrical Shell Method:
Disc Method:
= ∫(𝑎𝑟𝑒𝑎 𝑜𝑓 𝑐𝑖𝑟𝑐𝑙𝑒) 𝑑(𝑡ℎ𝑖𝑐𝑘𝑛𝑒𝑠𝑠)
𝑎
Washer Method - Rotation about
the x-axis:
Volume of
Revolution
𝑏
𝑉 = ∫ 𝜋 { [𝑓(𝑥)]2 − [𝑔(𝑥)]2 } 𝑑𝑥
𝑎
Cylinder Method - Rotation about
the y-axis:
𝑏
𝑉 = ∫ 2𝜋𝑥 𝑓(𝑥) 𝑑𝑥
𝑏
𝑎
= ∫(𝑐𝑖𝑟𝑐𝑢𝑚𝑓𝑒𝑟𝑒𝑛𝑐𝑒) (ℎ𝑖𝑔ℎ𝑡) 𝑑𝑥
𝑎
Copyright © 2011-2014 by Harold A. Toomey
10
.