Harold’s Calculus 3 Formulas

Harold’s Calculus 3 Formulas (Rectangular – Polar – Cylindrical – Spherical – Parametric – Vector – Matrix) Cheat Sheet 6 November 2025 Rectangular Polar/Cylindrical 2D: 𝑓 (𝑥 ) = 𝑦 (𝑥, 𝑦) or (𝑎, 𝑏) (𝑟 , 𝜃) or 𝑟 ∠ 𝜃 3D: 𝑓(𝑥, 𝑦) = 𝑧 (𝑥, 𝑦, 𝑧) Point 4D: 𝑓(𝑥, 𝑦, 𝑧) = 𝑤 (𝑥, 𝑦, 𝑧, 𝑤) Spherical Parametric Vector Matrix (𝜌 , 𝜃, 𝜙) Polar → Rect. 𝑥 = 𝑟 cos 𝜃 𝑦 = 𝑟 𝑠𝑖𝑛 𝜃 𝑧=𝑧 𝑦 𝑡𝑎𝑛 𝜃 = ( ) 𝑥 Rect. → Polar 2 2 2 𝑟 = 𝑥 +𝑦 𝑟 = √𝑥 2 + 𝑦 2 𝑦 𝜃 = 𝑡𝑎𝑛−1 ( ) 𝑥 𝑥 = 𝜌 𝑠𝑖𝑛 𝜙 𝑐𝑜𝑠 𝜃 𝑦 = 𝜌 sin 𝜙 𝑠𝑖𝑛 𝜃 𝑧 = 𝜌 𝑐𝑜𝑠 𝜙 𝜌2 = 𝑟 2 + 𝑧 2 𝜌 = 𝑥2 + 𝑦2 + 𝑧2 2 𝑦 𝑡𝑎𝑛 𝜃 = ( ) 𝑥 Point (𝑎, 𝑏) in Rectangular : 𝑥 (𝑡 ) = 𝑎 𝑦 (𝑡 ) = 𝑏 < 𝑎, 𝑏 > 𝑡 = 3𝑟𝑑 , variable, usually time, with 1 degree of freedom (df) ⃗ = 〈𝑥0 , 𝑦0 , 𝑧0 〉 𝒓 [𝑎] [𝑥 ] = [𝑏] ̂ ⃗ = 𝑥𝒊̂ + 𝑦𝒋̂ + 𝑧𝒌 𝒓 𝑧 𝜙 = cos −1 ( ) √𝑥 2 + 𝑦 2 + 𝑧 2 𝑧 𝜙 = cos −1 ( ) 𝜌 Slope-Intercept Form: 𝑦 = 𝑚𝑥 + 𝑏 Point-Slope Form: 𝑦 − 𝑦0 = 𝑚 (𝑥 − 𝑥0 ) Line Normal Form: 𝐴𝑥 + 𝐵𝑦 + 𝐶 = 0 𝑤ℎ𝑒𝑟𝑒 𝐴 𝑎𝑛𝑑 𝐵 ≠ 0 < 𝑥, 𝑦 > = < 𝑥0 , 𝑦0 > + 𝑡 < 𝑎, 𝑏 > < 𝑥, 𝑦 > = < 𝑥0 + 𝑎𝑡, 𝑦0 + 𝑏𝑡 > where < 𝑎, 𝑏 > = < 𝑥2 − 𝑥1 , 𝑦2 − 𝑦1 > Calculus Form: 𝑓(𝑥 ) = 𝑓 ′ (𝑎) 𝑥 + 𝑓(0) where 𝑚 = 𝑓’(𝑎) 𝑥(𝑡) = 𝑥0 + 𝑡𝑎 𝑦(𝑡) = 𝑦0 + 𝑡𝑏 Intercept Form: 𝑥 𝑦 + =1 𝑎 𝑏 𝑚= ⃗ = 𝒓 ⃗⃗⃗⃗𝟎 + 𝑡 𝒗 ⃗ 𝒓 = 〈𝑥0 , 𝑦0 , 𝑧0 〉 + 𝑡 〈𝑎, 𝑏, 𝑐〉 [𝑎 𝑥 𝑏] [𝑦] = [𝑐 ] 𝑎 [ 𝑐 𝑒 𝑏 𝑥 ] [𝑦 ] = [𝑓 ] 𝑑 ∆𝑦 𝑦2 − 𝑦1 𝑏 = = ∆𝑥 𝑥2 − 𝑥1 𝑎 3D: 𝑥 − 𝑥0 𝑦 − 𝑦0 𝑧 − 𝑧0 = = 𝑎 𝑏 𝑐 Copyright © 2011-2025 by Harold Toomey, WyzAnt Tutor 1 Rectangular Polar/Cylindrical Spherical Parametric Vector Matrix Point-Normal Form: 𝒏 ⦁ (𝒓 − 𝒓0 ) = 0 or 𝑛𝑥 (𝑥 − 𝑥0 ) + 𝑛𝑦 (𝑦 − 𝑦0 ) + 𝑛𝑧 (𝑧 − 𝑧0 ) = 0 Vector Form: 𝒓 = 𝒓0 + 𝑠𝒗 + 𝑡𝒘 Dot Product of Point-Normal Form: 𝑛𝑥 (𝑥 − 𝑥0 ) + 𝑛𝑦 (𝑦 − 𝑦0 ) + 𝑛𝑧 (𝑧 − 𝑧0 ) = 0 Plane where: 𝒏 = 〈𝑛𝑥 , 𝑛𝑦 , 𝑛𝑧 〉 = 〈𝑎, 𝑏, 𝑐〉 is the normal vector General Form: 𝐴𝑥 + 𝐵𝑦 + 𝐶𝑧 + 𝐷 = 0 Intercept Form: 𝑥 𝑦 𝑧 + + =1 𝑎 𝑏 𝑐 Copyright © 2011-2025 by Harold Toomey, WyzAnt Tutor (𝑟, 𝜃, 𝑘) (0 ≤ 𝑟 < ∞ ) (0 ≤ 𝜃 < 2𝜋) 𝑤ℎ𝑒𝑟𝑒 𝑟 𝑎𝑛𝑑 𝜃 𝑡𝑎𝑘𝑒 𝑜𝑛 𝑎𝑙𝑙 𝑣𝑎𝑙𝑢𝑒𝑠 𝑖𝑛 𝑡ℎ𝑒𝑖𝑟 𝑑𝑜𝑚𝑎𝑖𝑛s (𝜌, 𝜃, 𝑘) (0 ≤ 𝜌 < ∞ ) (0 ≤ 𝜃 < 2𝜋) 𝑤ℎ𝑒𝑟𝑒 𝜌 𝑎𝑛𝑑 𝜃 𝑡𝑎𝑘𝑒 𝑜𝑛 𝑎𝑙𝑙 𝑣𝑎𝑙𝑢𝑒𝑠 𝑖𝑛 𝑡ℎ𝑒𝑖𝑟 𝑑𝑜𝑚𝑎𝑠𝑖𝑛𝑠 Parametric Form: 𝑥 = 𝑥0 + 𝑠𝑢1 + 𝑡𝑣1 𝑦 = 𝑦0 + 𝑠𝑢2 + 𝑡𝑣2 𝑧 = 𝑧0 + 𝑠𝑢3 + 𝑡𝑣3 where: • (𝑥0 , 𝑦0 , 𝑧0 ) is a point on the plane. • 〈𝑢1 , 𝑢2 , 𝑢3 〉 and 〈𝑣1 , 𝑣2 , 𝑣3 〉 are direction vectors on the plane. • s and t are parameters that vary over all real numbers. where: • 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 Normal Vector: 𝒏 = 𝒗𝟏 × 𝒗𝟐 = 〈𝑛𝑥 , 𝑛𝑦 , 𝑛𝑧 〉 𝒊̂ 𝒏 = |𝑥2 − 𝑥1 𝑥3 − 𝑥1 𝒋̂ 𝑦2 − 𝑦1 𝑦3 − 𝑦1 ̂ 𝒌 𝑧2 − 𝑧1 | 𝑧3 − 𝑧1 Points on the plane: • 𝑃0 = (𝑥0 , 𝑦0 , 𝑧0 ) → 𝒓𝟎 = 〈𝑃0 〉 • 𝑃1 = (𝑥1 , 𝑦1 , 𝑧1 ) • 𝑃2 = (𝑥2 , 𝑦2 , 𝑧3 ) • 𝑃3 = (𝑥3 , 𝑦3 , 𝑧3 ) Vectors on the plane: • 𝒗𝟏 = 〈𝑃2 − 𝑃1 〉 = 〈𝑥2 − 𝑥1 , 𝑦2 − 𝑦1 , 𝑧2 − 𝑧1 〉 • 𝒗𝟐 = 〈𝑃3 − 𝑃1 〉 = 〈𝑥3 − 𝑥1 , 𝑦3 − 𝑦1 , 𝑧3 − 𝑧1 〉 2 Rectangular Polar/Cylindrical General Equation for All Conics: General Equation for All Conics: 𝐴𝑥 2 + 𝐵𝑥𝑦 + 𝐶𝑦 2 + 𝐷𝑥 + 𝐸𝑦 + 𝐹 =0 Vertical Axis of Symmetry: 𝑝 𝑟= 1 − 𝑒 𝑐𝑜𝑠 𝜃 where 𝐴=𝐵=𝐶=0 𝐴 = 𝐶 𝑎𝑛𝑑 𝐵 = 0 𝐴𝐶 > 0 𝑜𝑟 𝐵2 − 4𝐴𝐶 < 0 𝑃𝑎𝑟𝑎𝑏𝑜𝑙𝑎: 𝐴𝐶 = 0 or 𝐵2 − 4𝐴𝐶 = 0 𝐻𝑦𝑝𝑒𝑟𝑏𝑜𝑙𝑎: 𝐴𝐶 < 0 𝑜𝑟 𝐵2 − 4𝐴𝐶 > 0 Conics Parametric Vector Matrix Horizontal Axis of Symmetry: 𝑝 𝑟= 1 − 𝑒 𝑠𝑖𝑛 𝜃 𝐿𝑖𝑛𝑒: 𝐶𝑖𝑟𝑐𝑙𝑒: 𝐸𝑙𝑙𝑖𝑝𝑠𝑒: Note: If 𝐴 + 𝐶 = 0, then square hyperbola Spherical 𝑎 (1 − 𝑒 2 ) 0≤𝑒<1 𝑤ℎ𝑒𝑟𝑒 𝑝 = { 𝑓𝑜𝑟 { 𝑒 = 1 2𝑑 𝑒>1 𝑎 (𝑒 2 − 1) Quadratic Form: 𝐴 𝐵/2 𝑥 (𝑥 𝑦 ) ( ) (𝑦 ) 𝐵/2 𝐶 𝑥 + (𝐷 𝐸 ) ( 𝑦 ) + 𝐹 𝑝 = semi-latus rectum or the line segment running from the focus to the curve in a direction parallel to the directrix Rotation: If B ≠ 0, then rotate the 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 © 2011-2025 by Harold Toomey, WyzAnt Tutor Eccentricity: 𝐶𝑖𝑟𝑐𝑙𝑒 𝑒=0 𝐸𝑙𝑙𝑖𝑝𝑠𝑒 0<𝑒<1 𝑃𝑎𝑟𝑎𝑏𝑜𝑙𝑎 𝑒=1 𝐻𝑦𝑝𝑒𝑟𝑏𝑜𝑙𝑎 𝑒>1 =0 Matrix Form: 𝒙 𝑻 𝐴𝑄 𝒙 = 0 where 𝑥 𝒙 = (𝑦) 1 𝐴 𝐴𝑄 = (𝐵/2 𝐷/2 𝐵/2 𝐶 𝐸/2 𝐷/2 𝐸/2) 𝐹 3 Rectangular Polar/Cylindrical (𝑥 − ℎ)2 + (𝑦 − 𝑘)2 = 𝑟 2 Circle Parametric Centered at Origin: r = a (constant) 𝜃 = 𝜃 [0, 2𝜋] 𝑜𝑟 [0, 360°] 𝑥2 + 𝑦2 = 𝑟2 General Form: 2 𝐴𝑥 + 𝐵𝑥𝑦 + 𝐶𝑦 2 + 𝐷𝑥 + 𝐸𝑦 + 𝐹 =0 𝑤ℎ𝑒𝑟𝑒 𝐴 = 𝐶 𝑎𝑛𝑑 𝐵 = 0 Spherical Hint: Law of Cosines or 𝑟 = 𝑟0 cos(𝜃 − 𝜙) + √𝑎2 − 𝑟02 𝑠𝑖𝑛2 (𝜃 − 𝜙) Matrix 2D: |𝒙 − 𝒎| = 𝑟 Centered at (𝑟0 , 𝜙): 𝑟 2 + 𝑟02 − 2𝑟𝑟0 cos(𝜃 − 𝜙) = 𝑅2 Center: (ℎ, 𝑘) Vertices: NA Focus: (ℎ, 𝑘) Vector 𝜌 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝜃 = 𝜃 [0, 2𝜋] 𝜙 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 = 0 𝑥(𝑡) = 𝑟 𝑐𝑜𝑠(𝑡) + ℎ 𝑦(𝑡) = 𝑟 𝑠𝑖𝑛(𝑡) + 𝑘 [𝑡𝑚𝑖𝑛 , 𝑡𝑚𝑎𝑥 ] = [0, 2𝜋) Center: (ℎ, 𝑘) Focus: (ℎ, 𝑘) x = vector of the points of the circle m = vector to the center of the circle r = radius 𝐴 𝐵/2 𝑥 (𝑥 𝑦 ) ( ) (𝑦 ) 𝐵/2 𝐶 𝑥 + (𝐷 𝐸 ) ( 𝑦 ) + 𝐹 =0 3D: 𝒙 = 𝒄 + 𝑟 cos(𝑡) 𝒖 + 𝑟 sin(𝑡) 𝒗 𝒄 = (𝑐𝑥 , 𝑐𝑦 , 𝑐𝑧 ) center of the circle 𝒖, 𝒗 are two orthogonal unit vectors in the plane of the circle 𝑡 = [0, 2𝜋] where 𝐴 𝐵/2 )>0 det ( 𝐵/2 𝐶 𝐴=𝐶 𝐵=0 𝑥2 + 𝑦2 + 𝑧2 = 𝑟2 (𝑥 − ℎ)2 + (𝑦 − 𝑘)2 + (𝑧 − 𝑙)2 = 𝑟2 𝜌 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝜃 = 𝜃 [0, 2𝜋] 𝜙 = 𝜙 [0, 2𝜋] 𝐹𝑜𝑐𝑢𝑠 𝑎𝑛𝑑 𝑐𝑒𝑛𝑡𝑒𝑟: (ℎ, 𝑘, 𝑙) Sphere General Form: 𝐴𝑥 2 + 𝐵𝑦 2 + 𝐶𝑧 2 + 𝐷𝑥𝑦 + 𝐸𝑦𝑧 + 𝐹𝑥𝑧 + 𝐺𝑥 + 𝐻𝑦 + 𝐼𝑧 + 𝐽 = 0 𝑤ℎ𝑒𝑟𝑒 𝐴 = 𝐵 = 𝐶 > 0 Cylindrical to Rectangular: 𝑥 = 𝑟 𝑐𝑜𝑠 (𝜃) 𝑦 = 𝑟 𝑠𝑖𝑛 (𝜃) 𝑧=𝑧 Spherical to Rectangular: 𝑥 = 𝑟 𝑠𝑖𝑛 𝜃 𝑐𝑜𝑠 𝜙 𝑦 = 𝑟 𝑠𝑖𝑛 𝜃 𝑠𝑖𝑛 𝜙 𝑧 = 𝑟 𝑐𝑜𝑠 𝜃 Copyright © 2011-2025 by Harold Toomey, WyzAnt Tutor Rectangular to Cylindrical: 𝑟 = √𝑥 2 + 𝑦 2 Spherical to Cylindrical: 𝜌 = 𝑟 𝑠𝑖𝑛 (𝜃) 𝜙=𝜙 𝑧 = 𝑟 𝑐𝑜𝑠 (𝜃) Rectangular to Spherical: 𝑟 = √𝑥 2 + 𝑦 2 + 𝑧 2 𝑧 𝜃 = 𝑎𝑟𝑐𝑐𝑜𝑠 ( ) 𝑟 𝑦 𝜙 = 𝑎𝑟𝑐𝑡𝑎𝑛 ( ) 𝑥 Cylindrical to Spherical: 𝑟 = √𝜌2 + 𝑧 2 𝜌 𝑧 𝜃 = 𝑎𝑟𝑐𝑡𝑎𝑛 ( ) = 𝑎𝑟𝑐𝑐𝑜𝑠 ( ) 𝑧 𝑟 𝜙=𝜙 Rectangular: 𝑥 𝑦 𝒓≡[ ] 𝑧 Cylindrical: 𝑟 𝑐𝑜𝑠 (𝜃) 𝒓 ≡ [ 𝑟 𝑠𝑖𝑛 (𝜃) ] 𝑧 Spherical: 𝑟 𝑠𝑖𝑛 𝜃 𝑐𝑜𝑠 𝜙 𝒓 ≡ [ 𝑟 𝑠𝑖𝑛 𝜃 𝑠𝑖𝑛 𝜙 ] 𝑟 𝑐𝑜𝑠 𝜃 4 Rectangular 2 Polar/Cylindrical 𝑥 𝑦 + 2=1 2 𝑎 𝑏 2 Spherical Vertical Axis of Symmetry: 𝑎 (1 − 𝑒 2 ) 𝑟= 1 ± 𝑒 cos 𝜃 2 (𝑥 − ℎ) (𝑦 − 𝑘) + =1 2 𝑎 𝑏2 Vector Matrix 𝒙 = 𝒄 + 𝑎 cos(𝑡) 𝒖 + 𝑏 sin(𝑡) 𝒗 𝐴 𝐵/2 𝑥 (𝑥 𝑦 ) ( ) (𝑦 ) 𝐵/2 𝐶 𝑥(𝑡) = 𝑎 𝑐𝑜𝑠(𝑡) + ℎ 𝑦(𝑡) = 𝑏 𝑠𝑖𝑛(𝑡) + 𝑘 [𝑡𝑚𝑖𝑛 , 𝑡𝑚𝑎𝑥 ] = [0, 2𝜋] Horizontal Axis of Symmetry: 𝑎 (1 − 𝑒 2 ) 𝑟= 1 ± 𝑒 sin 𝜃 General Form: 𝐴𝑥 + 𝐵𝑥𝑦 + 𝐶𝑦 2 + 𝐷𝑥 + 𝐸𝑦 +𝐹 = 0 where 𝐵2 − 4𝐴𝐶 < 0 𝑜𝑟 𝐴𝐶 > 0 2 Center: (ℎ, 𝑘) 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 𝐸𝑐𝑐𝑒𝑛𝑡𝑟𝑖𝑐𝑖𝑡𝑦: 0 < 𝑒 < 1 Center: (ℎ, 𝑘) Vertices: (ℎ ± 𝑎, 𝑘) Co-Vertices: (ℎ, 𝑘 ± 𝑏) Foci: (ℎ ± 𝑐, 𝑘) Ellipse Parametric 2 𝑟 (𝜃 ) = Focus length, c, from center: 𝑐 2 = 𝑎2 − 𝑏 2 𝑎𝑏 √(𝑏 𝑐𝑜𝑠 𝜃)2 + (𝑎 𝑠𝑖𝑛 𝜃)2 relative to center (ℎ, 𝑘) 𝜃 = the angle between the x-axis and the major axis of the ellipse Eccentricity: 𝑐 √𝑎2 − 𝑏 2 𝑒= = 𝑎 𝑎 𝒄 = (𝑐𝑥 , 𝑐𝑦 , 𝑐𝑧 ) center of the ellipse 𝒖, 𝒗 are two orthogonal unit vectors in the plane of the ellipse 𝑡 = [0, 2𝜋𝑠] 𝑥 + (𝐷 𝐸 ) ( 𝑦 ) + 𝐹 =0 where 𝐴 𝐵/2 )>0 det ( 𝐵/2 𝐶 Interesting Note: The sum of the distances from each focus to a point on the curve is constant. |𝑑1 + 𝑑2 | = 𝑘 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 2 Ellipsoid (𝑥 − ℎ)2 (𝑦 − 𝑘)2 (𝑧 − 𝑙)2 + + 𝑎2 𝑏2 𝑐2 =1 𝑟 2 𝑐𝑜𝑠 2 𝜃 𝑟 2 𝑠𝑖𝑛2 𝜃 𝑧 2 + + 2=1 𝑎2 𝑏2 𝑐 2 2 𝑟 𝑐𝑜𝑠 𝜃 𝑠𝑖𝑛 𝜙 𝑎2 2 𝑟 𝑠𝑖𝑛2 𝜃 𝑠𝑖𝑛2 𝜙 + 𝑏2 2 𝑟 𝑐𝑜𝑠 2 𝜙 + =1 𝑐2 𝑥(𝑡, 𝑢) = 𝑎 𝑐𝑜𝑠(𝑡) 𝑐𝑜𝑠(𝑢) + ℎ 𝑦(𝑡, 𝑢) = 𝑏 𝑐𝑜𝑠(𝑡) 𝑠𝑖𝑛(𝑢) + 𝑘 𝑧(𝑡, 𝑢) = 𝑐 𝑠𝑖𝑛(𝑡) + 𝑙 𝜋 𝜋 [𝑡𝑚𝑖𝑛 , 𝑡𝑚𝑎𝑥 ] = [− , ] 2 2 [𝑢𝑚𝑖𝑛 , 𝑢𝑚𝑎𝑥 ] = [−𝜋, 𝜋] (𝒙 − 𝒗)𝑇 𝐴−1 (𝒙 − 𝒗) = 1 Centered at vector 𝒗 𝐶𝑒𝑛𝑡𝑒𝑟: (ℎ, 𝑘, 𝑙 ) Copyright © 2011-2025 by Harold Toomey, WyzAnt Tutor 5 Rectangular 2 Polar/Cylindrical Spherical Parametric Vector Matrix 2 𝑥 𝑦 − 2=1 2 𝑎 𝑏 (𝑥 − ℎ )2 (𝑦 − 𝑘 ) 2 − =1 𝑎2 𝑏2 Vertical Axis of Symmetry: 𝑎 (𝑒 2 − 1) 𝑟= 1 ± 𝑒 cos 𝜃 General Form: 𝐴𝑥 + 𝐵𝑥𝑦 + 𝐶𝑦 2 + 𝐷𝑥 + 𝐸𝑦 + 𝐹 =0 where 𝐵2 − 4𝐴𝐶 > 0 𝑜𝑟 𝐴𝐶 < 0 Horizontal Axis of Symmetry: 𝑎 (𝑒 2 − 1) 𝑟= 1 ± 𝑒 sin 𝜃 2 𝐸𝑐𝑐𝑒𝑛𝑡𝑟𝑖𝑐𝑖𝑡𝑦: 𝑒 > 1 𝑐 √𝑎2 + 𝑏 2 𝑤ℎ𝑒𝑟𝑒 𝑒 = = 𝑎 𝑎 = sec 𝜃 > 1 relative to center (ℎ, 𝑘) If 𝐴 + 𝐶 = 0, square hyperbola Center: (ℎ, 𝑘) Vertices: (ℎ ± 𝑎, 𝑘) Foci: (ℎ ± 𝑐, 𝑘) Hyperbola 1 1 −𝑐𝑜𝑠 −1 (− ) < 𝜃 < 𝑐𝑜𝑠 −1 (− ) 𝑒 𝑒 Focus length, c, from center: 𝑐 2 = 𝑎2 + 𝑏 2 Eccentricity: 𝑐 √𝑎2 + 𝑏 2 𝑒= = = sec 𝜃 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 Left-Right Opening Hyperbola: 𝑥(𝑡) = 𝑎 𝑠𝑒𝑐(𝑡) + ℎ 𝑦(𝑡) = 𝑏 𝑡𝑎𝑛(𝑡) + 𝑘 [𝑡𝑚𝑖𝑛 , 𝑡𝑚𝑎𝑥 ] = [−𝑐, 𝑐] Vertex: (h, k) Alternate Form: 𝑥(𝑡) = ±𝑎 𝑐𝑜𝑠ℎ(𝑡) + ℎ 𝑦(𝑡) = 𝑏 𝑠𝑖𝑛ℎ(𝑡) + 𝑘 Up-Down Opening Hyperbola: 𝑥(𝑡) = 𝑎 𝑡𝑎𝑛(𝑡) + ℎ 𝑦(𝑡) = 𝑏 𝑠𝑒𝑐(𝑡) + 𝑘 [𝑡𝑚𝑖𝑛 , 𝑡𝑚𝑎𝑥 ] = [−𝑐, 𝑐] Vertex: (h, k) Alternate Form: 𝑥(𝑡) = 𝑎 𝑠𝑖𝑛ℎ(𝑡) + ℎ 𝑦(𝑡) = ±𝑏 𝑐𝑜𝑠ℎ(𝑡) + 𝑘 Interesting Note: The difference between the distances from each focus to a point on the curve is constant. |𝑑1 − 𝑑2 | = 𝑘 p = semi-latus rectum or the line segment running from the focus to the curve in the directions 𝜋 𝜃=± 𝐴 𝐵/2 𝑥 (𝑥 𝑦 ) ( ) (𝑦 ) 𝐵/2 𝐶 𝑥 + (𝐷 𝐸 ) ( 𝑦 ) + 𝐹 =0 where 𝐴 𝐵/2 )<0 det ( 𝐵/2 𝐶 General Form: 𝑥(𝑡) = 𝐴𝑡 2 + 𝐵𝑡 + 𝐶 𝑦(𝑡) = 𝐷𝑡 2 + 𝐸𝑡 + 𝐹 where A and D have different signs 2 New = (x’, y’), Old = (x, y) rotates through angle 𝜃 from x-axis (𝑥 − ℎ )2 (𝑦 − 𝑘 )2 (𝑧 − 𝑙 )2 + − 𝑎2 𝑏2 𝑐2 =1 Hyperboloid − (𝑥 − ℎ )2 (𝑦 − 𝑘 ) 2 (𝑧 − 𝑙 )2 − + 𝑎2 𝑏2 𝑐2 =1 Copyright © 2011-2025 by Harold Toomey, WyzAnt Tutor 6 Rectangular Polar/Cylindrical Spherical Parametric Vector Matrix 2 𝑦 = 𝑎𝑥 + 𝑏𝑥 + 𝑐 𝑦 = ( 𝑥 − ℎ )2 + 𝑘 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 Parabola 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 General Form: 2 𝐴𝑥 + 𝐵𝑥𝑦 + 𝐶𝑦 2 + 𝐷𝑥 + 𝐸𝑦 + 𝐹 =0 where 𝐵2 − 4𝐴𝐶 = 0 or 𝐴𝐶 = 0 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 Vertical Axis of Symmetry: 𝑥 (𝑡) = 2𝑝𝑡 + ℎ 𝑦(𝑡) = 𝑝𝑡 2 + 𝑘 (opens upwards) 𝑦(𝑡) = −𝑝𝑡 2 − 𝑘 (opens downwards) [𝑡𝑚𝑖𝑛 , 𝑡𝑚𝑎𝑥 ] = [−𝑐, 𝑐] Vertex: (ℎ, 𝑘) Vertical Axis of Symmetry: 𝑒𝑑 𝑟= 1 ± 𝑒 cos 𝜃 Horizontal Axis of Symmetry: 𝑒𝑑 𝑟= 1 ± 𝑒 sin 𝜃 Horizontal Axis of Symmetry: 𝑦(𝑡) = 2𝑝𝑡 + 𝑘 𝑥 (𝑡) = 𝑝𝑡 2 + ℎ (opens to the right) 𝑥 (𝑡) = −𝑝𝑡 2 − ℎ (opens to the left) [𝑡𝑚𝑖𝑛 , 𝑡𝑚𝑎𝑥 ] = [−𝑐, 𝑐] Vertex: (ℎ, 𝑘) 𝐸𝑐𝑐𝑒𝑛𝑡𝑟𝑖𝑐𝑖𝑡𝑦: 𝑒 = 1 and 𝑑 = 2𝑝 Interesting Note: The distances from a point on the curve to the focus is the same as to the directrix. Projectile Motion: 1 𝑥 (𝑡) = 𝑥0 + 𝑣𝑥 𝑡 + ( ) 𝑎𝑥 𝑡 2 2 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 𝐴 𝐵/2 𝑥 (𝑥 𝑦 ) ( ) (𝑦 ) 𝐵/2 𝐶 𝑥 + (𝐷 𝐸 ) ( 𝑦 ) + 𝐹 =0 where 𝐴 𝐵/2 )=0 det ( 𝐵/2 𝐶 General Form: 𝑥 = 𝐴𝑡 2 + 𝐵𝑡 + 𝐶 𝑦 = 𝐿𝑡 2 + 𝑀𝑡 + 𝑁 where A and L have the same sign New = (x’, y’), Old = (x, y) rotates through angle 𝜃 from x-axis Paraboloid (𝑥 − ℎ)2 (𝑦 − 𝑘)2 (𝑧 − 𝑙)2 + = 𝑎2 𝑏2 𝑐2 Copyright © 2011-2025 by Harold Toomey, WyzAnt Tutor 7 Rectangular 2nd Derivative Parametric Vector Matrix 𝑥→𝑐 𝑓(𝑥 + ℎ) − 𝑓(𝑥) ℎ→0 ℎ ( ) 𝑓 𝑥 − 𝑓(𝑐) 𝑓 ′ (𝑐 ) = lim 𝑥→𝑐 𝑥−𝑐 𝑑𝑦 ′( ) 𝑓 𝑥 = = 𝑦′ = 𝐷𝑥 𝑑𝑥 𝑓 ′ (𝑥 ) = lim 1 Derivative Spherical lim 𝑓(𝑥) = 𝐿 Limit st Polar/Cylindrical 𝑑 𝑑𝑦 𝑑2 𝑦 ′′ ( ) ( )= 𝑓 𝑥 = = 𝑦′′ 𝑑𝑥 𝑑𝑥 𝑑𝑥 2 = 𝐷𝑥𝑥 𝑑𝑦 𝑑𝑟 𝑑𝑦 𝑑𝜃 𝑑𝜃 𝑠𝑖𝑛 𝜃 + 𝑟 𝑐𝑜𝑠 𝜃 = = 𝑑𝑟 𝑑𝑥 𝑑𝑥 𝑐𝑜𝑠 𝜃 − 𝑟 𝑠𝑖𝑛 𝜃 𝑑𝜃 𝑑𝜃 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 𝑑 (𝒓 ⃗)= 𝒓 ⃗ ′ 𝑑𝑡 𝑑𝑦 𝑑𝑦 = 𝑑𝑡 , 𝑑𝑥 𝑑𝑥 𝑑𝑡 𝑑𝑥 𝑝𝑟𝑜𝑣𝑖𝑑𝑒𝑑 ≠0 𝑑𝑡 𝑑𝑦 𝑑 𝑑𝑡 ( ) 𝑑 𝑑𝑦 𝑑𝑡 𝑑𝑥 ( ) 𝑑2 𝑦 𝑑 𝑑𝑦 𝑑𝑡 𝑑𝑥 𝑑𝑡 ( )= = = 𝑑𝑥 𝑑𝑥 𝑑𝑥 2 𝑑𝑥 𝑑𝑥 𝑑𝑡 𝑑𝑡 Riemann Sum: Unit tangent vector ⃗𝑇(𝑡) ⃗ ′(𝑡) 𝒓 ⃗ ′ (𝑡 ) = 𝑤ℎ𝑒𝑟𝑒 𝒓 ‖𝒓 ⃗ ′(𝑡)‖ ⃗ ≠0 Unit normal vector ⃗𝑁(𝑡) ⃗ ′(𝑡) 𝑻 ⃗ ′ (𝑡 ) = 𝑤ℎ𝑒𝑟𝑒 𝑻 ⃗ ‖𝑻′(𝑡)‖ ⃗ ≠0 𝜕𝑓 𝜕𝑓 1 𝜕𝑓 = 𝐞𝑅 + 𝐞 𝜕𝒓 𝜕𝑅 𝑅 𝜕𝜃 𝜃 1 𝜕𝑓 + 𝐞 𝑅 sin (𝜃) 𝜕𝜑 𝜑 𝑛 𝑆=∑ 𝑓(𝑦𝑖 )(𝑥𝑖 − 𝑥𝑖−1 ) 𝑖−1 Left Sum: 1 𝑓(𝑎) + 𝑓 (𝑎 + ) + 1 𝑛 ] 𝑆 = ( )[ 2 1 𝑛 𝑓 (𝑎 + ) + ⋯ + 𝑓(𝑏 − ) 𝑛 𝑛 Fundamental Theorem of Calculus: Integral 𝑏 Middle Sum: 1 1 3 𝑆 = ( ) [𝑓 (𝑎 + ) + 𝑓 (𝑎 + ) + ⋯ 𝑛 2𝑛 2𝑛 1 + 𝑓(𝑏 − )] 2𝑛 𝐹 (𝑥 ) = ∫ 𝑓(𝑥 ) 𝑑𝑥 = 𝐹 (𝑏) − 𝐹(𝑎) 𝑎 𝑏 𝑏 𝑏 𝑏 ∫ ⃗𝒓(𝑡) 𝑑𝑡 = 〈∫ 𝑓(𝑡)𝑑𝑡 , ∫ 𝑔(𝑡)𝑑𝑡 , ∫ ℎ(𝑡)𝑑𝑡〉 𝑎 𝑎 𝑎 𝑎 Right Sum: 1 1 2 𝑆 = ( ) [𝑓 (𝑎 + ) + 𝑓 (𝑎 + ) + ⋯ 𝑛 𝑛 𝑛 + 𝑓(𝑏)] 𝑏 𝑑(𝑦) Double Integral ∫ ∫ 𝑓(𝑥, 𝑦) 𝑑𝑥 𝑑𝑦 𝑆𝑎𝑚𝑒 𝑎𝑠 𝑟𝑒𝑐𝑡𝑎𝑛𝑔𝑢𝑙𝑎𝑟, 𝑏𝑢𝑡 𝑓(𝑥, 𝑦) ⟶ 𝑓(𝜌 cos 𝜙 , 𝜌 sin 𝜙) 𝑎 𝑐(𝑦) Copyright © 2011-2025 by Harold Toomey, WyzAnt Tutor 8 Rectangular Polar/Cylindrical 𝑏 𝑑(𝑧) 𝑔(𝑦,𝑧) Triple Integral ∫ ∫ 𝑓(𝑥, 𝑦, 𝑧) 𝑑𝑥 𝑑𝑦 𝑑𝑧 ∫ if 𝑦 = 𝑠𝑖𝑛 𝜃 if 𝑦 = 𝑐𝑜𝑠 𝜃 if 𝑦 = 𝑡𝑎𝑛 𝜃 𝑡ℎ𝑒𝑛 𝜃 = 𝑠𝑖𝑛−1 𝑦 𝑡ℎ𝑒𝑛 𝜃 = 𝑐𝑜𝑠 −1 𝑦 𝑡ℎ𝑒𝑛 𝜃 = 𝑡𝑎𝑛−1 𝑦 𝑆𝑎𝑚𝑒 𝑎𝑠 𝑟𝑒𝑐𝑡𝑎𝑛𝑔𝑢𝑙𝑎𝑟, 𝑏𝑢𝑡 𝑓(𝑥, 𝑦, 𝑧) ⟶ 𝑓(𝜌 cos 𝜃 sin 𝜙, 𝜌 sin 𝜃 sin 𝜙 , 𝜌 cos 𝜙) or 𝜃 = arcsin 𝑦 or 𝜃 = arccos 𝑦 or 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 or 𝜃 = 𝑎𝑟𝑐𝑐𝑠𝑐 𝑦 or 𝜃 = 𝑎𝑟𝑐𝑠𝑒𝑐 𝑦 or 𝜃 = 𝑎𝑟𝑐𝑐𝑜𝑡 𝑦 𝑆𝑎𝑚𝑒 𝑎𝑠 𝑟𝑒𝑐𝑡𝑎𝑛𝑔𝑢𝑙𝑎𝑟, 𝑏𝑢𝑡 𝑓(𝑥, 𝑦, 𝑧) ⟶ 𝑓(𝜌 cos 𝜙 , 𝜌 sin 𝜙 , 𝑧) 𝑎 𝑐(𝑧) 𝑒(𝑦,𝑧) 𝐼𝑓 𝑓(𝑥 ) = 𝑦, 𝑡ℎ𝑒𝑛 𝑓 Inverse Functions −1 (𝑦) = 𝑥 Inverse Function Theorem: 1 𝑓 −1 (𝑓 ′ (𝑎)) = ′ 𝑓 (𝑎 ) Spherical Parametric Vector Matrix 𝑏 Rectangular 2D: 𝐿 = ∫ √1 + [𝑓 ′ (𝑥 )]2 𝑑𝑥 𝛽 Polar: 𝑎 𝑑𝑟 2 2 √ 𝐿 = ∫ 𝑟 + ( ) 𝑑𝜃 𝑑𝜃 Proof: ∆𝑠 = √(𝑥 − 𝑥0 )2 + (𝑦 − 𝑦0 )2 ∆𝑠 = √(∆𝑥)2 + (∆𝑦)2 𝑑𝑠 = √𝑑𝑥 2 + 𝑑𝑦 2 𝑑𝑠 = √𝑑𝑥 2 + 𝑑𝑠 = √𝑑𝑥 2 Where 𝑟 = 𝑓(𝜃) 𝑑𝑥 2 ) 𝑑𝑥 2 𝑑𝑦 2 2 2 √ ( ) 𝑑𝑥 𝑑𝑠 = 𝑑𝑥 + 𝑑𝑥 𝑑𝑦 2 (1 + ( ) ) 𝑑𝑥 𝛼 Circle: 𝐿 = 𝑠 = 𝑟𝜃 𝑑𝑥 2 𝑑𝑦 2 ) + ( ) 𝑑𝑡 𝑑𝑡 𝑑𝑡 Rectangular 3D: C = πd = 2πr 𝑑𝑦 2 ( Arc Length 𝐿 = ∫ √( 𝛽 𝑑𝑥 2 𝑑𝑦 2 𝑑𝑧 2 √ 𝐿 = ∫ ( ) + ( ) + ( ) 𝑑𝑡 𝑑𝑡 𝑑𝑡 𝑑𝑡 ⃗ ′(𝑡)‖ 𝑑𝑡 𝐿 = ∫ ‖𝒓 𝑡2 Proof: 𝐿 = (𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑐𝑖𝑟𝑐𝑢𝑚𝑓𝑒𝑟𝑒𝑛𝑐𝑒) ∙ 𝜋 ∙ (𝑑𝑖𝑎𝑚𝑒𝑡𝑒𝑟) 𝑑𝑟 2 𝑑𝜃 2 𝑑𝑧 2 𝐿 = ∫ √( ) + 𝑟 2 ( ) + ( ) 𝑑𝑡 𝑑𝑡 𝑑𝑡 𝑑𝑡 𝜃 𝐿 = ( ) 𝜋 (2𝑟) = 𝑟𝜃 2𝜋 Spherical: 𝐿= ∫ √( 𝑡1 𝑑𝜌 2 𝑑𝜃 2 𝑑𝜑 2 ) + 𝜌2 𝑠𝑖𝑛2 𝜑 ( ) + 𝜌2 ( ) 𝑑𝑡 𝑑𝑡 𝑑𝑡 𝑑𝑡 (𝑧 ′′ 𝑦 ′ |𝑦 ′′ | (1 + 3 𝑦 ′2 ) ⁄2 𝜅(𝜃) = for r(𝜃) 3⁄ 2 √(𝑥 ′′ 𝑧 ′ 𝜅= (𝑥 ′2 + 𝑦 ′2 + 𝜅= 3 𝑧 ′2 ) ⁄2 where f(t) = (x(t), y(t), z(t)) Copyright © 2011-2025 by Harold Toomey, WyzAnt Tutor ⃗ 𝑑𝑇 𝜅= | | 𝑑𝑠 𝑦 ′′ 𝑧 ′ )2 − + ′′ ′ 2 −𝑧 𝑥 ) + (𝑦 ′′ 𝑥 − 𝑥 ′′ 𝑦 ′ )2 |𝑟 2 + 2𝑟 ′2 − 𝑟𝑟 ′′ | (𝑟 2 + 𝑟 ′2 ) 𝑡 ⃗ ′(𝑢)‖ 𝑑𝑢 s(t) = ∫0 ‖𝒓 𝑡1 𝑡2 𝐿 = ∫ 𝑑𝑠 Curvature 𝑎 Cylindrical: 𝑑𝑦 2 𝑑𝑠 = √1 + ( ) 𝑑𝑥 𝑑𝑥 𝜅= 𝑏 𝛼 𝜅= ⃗ ′(𝑡)‖ ‖𝑇 ‖𝑟′(𝑡)‖ (See Wikipedia: Curvature) ‖𝑟 ′ (𝑡) × 𝑟′′(𝑡)‖ ‖𝑟′(𝑡)‖3 9 Rectangular Polar/Cylindrical Spherical Parametric Vector Matrix 𝑎2 +𝑏 2 Perimeter Area Square: Rectangle: Triangle: Circle: Ellipse: P = 4s P = 2l + 2w P=a+b+c C = πd = 2πr 𝐶 ≈ 𝜋(𝑎 + 𝑏) Square: Rectangle: Rhombus: Parallelogram: A = s² A = lw A = ½ ab A = Bh Trapezoid: 𝐴= 2 𝑑1 𝑑2 𝜋 2 𝐶 ≈ 𝜋 [3(𝑎 + 𝑏) − √(3𝑎 + 𝑏)(𝑎 + 3𝑏)] 3ℎ ) 𝐶 ≈ 𝜋 (𝑎 + 𝑏) (1 + 10 + √4 − 3ℎ Ellipse: 𝐶 = 4𝑎 ∫02 √1 − 𝑘 2 𝑠𝑖𝑛2 𝜃 𝑑𝜃 ℎ= (𝑎 − 𝑏 )2 𝑏2 2 & 𝑘 = (1 − ) (𝑎 + 𝑏 )2 𝑎2 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) ℎ Kite: 𝐴 = 2 Triangle: A = ½ Bh Triangle: A = ½ ab sin(C) Triangle using Heron’s Formula: 𝐴 = √𝑠(𝑠 − 𝑎)(𝑠 − 𝑏)(𝑠 − 𝑐 ) 𝑎+𝑏+𝑐 𝑤ℎ𝑒𝑟𝑒 𝑠 = 2 Equilateral Triangle: 𝐴 = ¼√3𝑠 2 Frustum: Circle: Circular Sector: Ellipse: Lateral Surface Area (𝐵1 + 𝐵2 ) Ellipse: 𝐶 ≈ 2𝜋√ Cylinder: Cone: 1 𝐵1 +𝐵2 )ℎ 𝐴= ( 3 2 A = πr² A = ½ r²𝜃 A = πab Simplified: 𝛽 𝐴 = ∫ 𝑦 (𝑡 ) 𝛼 𝜕𝒓 𝜕𝒓 𝐴 = ∬ | × | 𝑑𝑢 𝑑𝑣 𝜕𝑢 𝜕𝑣 𝐷 Proof: 𝑏 ∫ 𝑓(𝑥 ) 𝑑𝑥 Area of a sector where arc length 𝑠 = 𝑟𝜃: 1 𝐴 = ∫ 𝑠 𝑑𝑟 = ∫ 𝑟𝜃 𝑑𝑟 = 𝑟 2 𝜃 2 SA = 2πrh SA = πrl 𝑏 𝑎 y = f(x) = g(t) For rotation about the x-axis: 𝑑𝑓(𝑡) 𝑑𝑡 = 𝑓’(𝑡) 𝑑𝑡 𝑑𝑡 For rotation about the x-axis: 𝑆𝐴 = ∫ 2𝜋𝑦 𝑑𝑠 𝑆𝐴 = ∫ 2𝜋𝑦 𝑑𝑠 For rotation about the y-axis: For rotation about the y-axis: 𝑑𝑥 = Sphere: 𝑆𝐴 = 4𝜋𝑟² 𝑆𝐴 = ∫ 2𝜋𝑥 𝑑𝑠 𝑆𝐴 = ∫ 2𝜋𝑥 𝑑𝑠 𝑆𝐴 = 2𝜋 ∫ 𝑓 (𝑥 ) √1 + [𝑓 ′ (𝑥 )]2 𝑑𝑥 𝑎 𝑑𝑟 2 ) 𝑑𝜃 𝑑𝜃 𝑟 = 𝑓 (𝜃 ), 𝛼≤ 𝜃≤𝛽 Ellipsoid: SA ≈ 𝑑𝑥 2 𝑑𝑦 2 √ 𝑑𝑠 = ( ) + ( ) 𝑑𝑡 𝑑𝑡 𝑑𝑡 𝑖𝑓 𝑥 = 𝑓(𝑡), 𝑦 = 𝑔(𝑡), 𝛼 ≤ 𝑡 ≤ 𝛽 𝑑𝑠 = √𝑟 2 + ( Total Surface Area 𝑑𝑥(𝑡) 𝑑𝑡 𝑑𝑡 Cube: SA = 6s² Rectangular Box: SA = 2lw + 2wh + 2hl Regular Tetrahedron: SA = 2bh Cylinder: SA = 2πr (r + h) Cone: SA = πr² + πrl = πr (r + l) Sphere: SA = 4πr² Copyright © 2011-2025 by Harold Toomey, WyzAnt Tutor 1⁄ 𝑎𝑝 𝑏 𝑝 + 𝑎𝑝 𝑐 𝑝 + 𝑏 𝑝 𝑐 𝑝 𝑝 4𝜋 ( ) 3 Where p ≈ 1.6075, |𝑅𝑒𝑙𝑎𝑡𝑖𝑣𝑒 𝐸𝑟𝑟𝑜𝑟| ≤ 1.061% (Knud Thomsen’s Formula) Ellipsoid: S = where 10 Rectangular Polar/Cylindrical For revolution about the x-axis: For revolution about the x-axis: 𝑏 𝑑𝑦 2 √ 𝐴 = 2𝜋 ∫ 𝑓(𝑥) 1 + ( ) 𝑑𝑥 𝑑𝑥 Surface of Revolution 𝑎 Parametric Vector Matrix For revolution about the x-axis: 𝛽 𝑏 𝑑𝑟 2 2 √ 𝐴 = 2𝜋 ∫ 𝑟 𝑐𝑜𝑠 𝜃 𝑟 + ( ) 𝑑𝜃 𝑑𝜃 𝑑𝑥 2 𝑑𝑦 2 √ 𝐴𝑥 = 2𝜋 ∫ 𝑦(𝑡) ( ) + ( ) 𝑑𝑡 𝑑𝑡 𝑑𝑡 𝛼 𝑎 Sphere: S = 4πr² For revolution about the y-axis: For revolution about the y-axis: 𝑏 2 𝑑𝑥 𝐴 = 2𝜋 ∫ 𝑥 √1 + ( ) 𝑑𝑦 𝑑𝑦 𝑎 Sphere: = 𝑠³ = 𝑙𝑤ℎ = 𝜋𝑟²ℎ = 𝐵ℎ = ⅓ 𝐵ℎ = ⅓ 𝐵ℎ = ⅓ 𝑙𝑤ℎ 𝑉 = ⅓ 𝐵ℎ = ⅓ 𝜋𝑟²ℎ 4 𝑉 = 𝜋𝑟 3 Ellipsoid: 𝑉= Cube: Rectangular Prism: Cylinder: Triangular Prism: Tetrahedron: Pyramid: Volume Spherical Cone: 3 𝑏 𝑑𝑟 2 𝐴 = 2𝜋 ∫ 𝑟 𝑠𝑖𝑛 𝜃 √𝑟 2 + ( ) 𝑑𝜃 𝑑𝜃 𝐴𝑦 = 2𝜋 ∫ 𝑥 (𝑡) √( 𝛼 𝑉 𝑉 𝑉 𝑉 𝑉 𝑉 3 4 For revolution about the y-axis: 𝛽 ∫∫∫ 𝑓(𝑟 𝑐𝑜𝑠 𝜃 , 𝑟 𝑠𝑖𝑛 𝜃, 𝑧)𝑟 𝑑𝑧 𝑑𝑟 𝑑𝜃 𝑎 𝜌 𝑠𝑖𝑛 𝜑 𝑐𝑜𝑠 𝜃 , ∫ ∫ ∫ 𝑓 (𝜌 𝑠𝑖𝑛 𝜑 sin 𝜃 ,) 𝜌 𝑐𝑜𝑠 𝜑 … 𝜌2 𝑠𝑖𝑛 𝜑 𝑑𝜌 𝑑𝜑 𝑑𝜃 𝑑𝑥 2 𝑑𝑦 2 ) + ( ) 𝑑𝑡 𝑑𝑡 𝑑𝑡 Ellipsoid: 4 𝑉 = 𝜋√𝑑𝑒𝑡(𝐴−1 ) 3 𝜋𝑎𝑏𝑐 ∫ ∫ ∫ 𝑓 (𝑥, 𝑦, 𝑧) 𝑑𝑥 𝑑𝑦 𝑑𝑧 Copyright © 2011-2025 by Harold Toomey, WyzAnt Tutor 11 Rectangular Polar/Cylindrical Spherical Parametric Vector Matrix Disk Method 𝑉 𝑏 = ∫(𝑎𝑟𝑒𝑎 𝑜𝑓 𝑐𝑖𝑟𝑐𝑙𝑒) 𝑑 (𝑡ℎ𝑖𝑐𝑘𝑛𝑒𝑠𝑠) 𝑎 Rotation about the x-axis: 𝑏 𝑉 = ∫ 𝜋 [𝑓 (𝑥 )]2 𝑑𝑥 𝑎 Rotation about the y-axis: 𝑑 𝑉 = ∫ 𝜋𝑥 2 𝑑𝑦 𝑐 Washer Method Rotation about the x-axis: 𝑏 Volume of Revolution 𝑉 = ∫ 𝜋 { [𝑓(𝑥)]2 − [𝑔(𝑥 )]2 } 𝑑𝑥 𝑉 = 𝑉𝑂𝑢𝑡𝑒𝑟 𝐷𝑖𝑠𝑘 − 𝑉𝐼𝑛𝑛𝑒𝑟 𝐷𝑖𝑠𝑘 𝑎 Shell Method 𝑉 𝑏 = ∫(𝑐𝑖𝑟𝑐𝑢𝑚𝑓𝑒𝑟𝑒𝑛𝑐𝑒) (ℎ𝑖𝑔ℎ𝑡) 𝑑𝑥 𝑎 Rotation about the y-axis: 𝑏 𝑉 = ∫ 2𝜋𝑥 𝑓(𝑥 ) 𝑑𝑥 𝑎 Rotation about the x-axis: 𝑑 𝑉 = ∫ 2𝜋𝑦 𝑔(𝑦) 𝑑𝑦 𝑐 Copyright © 2011-2025 by Harold Toomey, WyzAnt Tutor 12 Rectangular 𝑁 𝑎 𝐼 = ∑ 𝑚𝑖 𝑟𝑖2 = ∫ 𝑚 𝑟 2 𝑑𝑟 Moment of Inertia 0 𝑖=1 Polar/Cylindrical Parametric 𝐴 𝑹= 1 ∑ 𝑚𝑖 𝑟𝑖 𝑀 𝑖=1 where 𝑀 = ∑𝑁 𝑖=1 𝑚𝑖 1D for Discrete: 𝑚1 𝑥1 + 𝑚2 𝑥2 𝑥𝑐𝑚 = 𝑚1 + 𝑚2 𝑉 3D for Discrete: 2D for Discrete: 𝑀𝑦 = ∑ 𝑚𝑖 𝑥𝑖 𝑖=1 𝑁 𝑀𝑥 = ∑ 𝑚𝑖 𝑦𝑖 𝑁 𝑥𝑐𝑚 1 = 𝑥̅ = ∑ 𝑚𝑖 𝑥𝑖 𝑀 𝑦𝑐𝑚 1 = 𝑦̅ = ∑ 𝑚𝑖 𝑦𝑖 𝑀 𝑖=1 𝑁 𝑖=1 𝑁 𝑖=1 𝑥̅ = 𝑀𝑦 𝑀𝑥 , 𝑦̅ = 𝑀 𝑀 Matrix = ∭ 𝜌(𝒓) 𝑑(𝒓)2 𝑑𝑉(𝒓) (see Wikipedia: Moment of Inertia) 𝐴 𝑁 Vector 𝐼 𝐽 = ∫ 𝑟 2 𝑑𝐴 = ∫(𝑥 2 + 𝑦 2 ) 𝑑𝐴 𝐽 = 𝐼𝑥 + 𝐼𝑦 𝑁 Center of Mass Spherical 𝑧𝑐𝑚 = 𝑧̅ = 1 ∑ 𝑚𝑖 𝑧𝑖 𝑀 𝑖=1 3D for Continuous: 1 𝑀 ∫ 𝑥 𝑑𝑚 𝑥̅ = 𝑀 0 1 𝑀 ∫ 𝑦 𝑑𝑚 𝑦̅ = 𝑀 0 1 𝑀 ∫ 𝑧 𝑑𝑚 𝑧̅ = 𝑀 0 𝑀 where 𝑀 = ∫0 𝑑𝑚 and 𝑑𝑚 = 𝜌 𝑑𝑧 𝑑𝑦 𝑑𝑥 𝑹= 1 ∫ 𝒓 𝑑𝑚 𝑀 1 ∭ 𝜌(𝒓) 𝒓 𝑑𝑉 𝑀 𝑹= 𝑉 Where 𝒓 is distance from the axis of rotation, not origin. (∇𝑓(x)) • 𝒗 = 𝐷𝒗 𝑓(𝑥) ∇𝑓 = Gradient 𝜕𝑓 𝜕𝑓 𝜕𝑓 𝒊+ 𝒋+ 𝒌 𝜕𝑥 𝜕𝑦 𝜕𝑧 ∇𝑓(ρ, ϕ, z) = ∇𝑓(r, θ, ϕ) = 𝜕𝑓 1 𝜕𝑓 𝜕𝑓 𝒆ρ + 𝒆ϕ + 𝒆 𝜕ρ ρ 𝜕ϕ 𝜕𝑧 z 𝜕𝑓 1 𝜕𝑓 𝟏 𝜕𝑓 𝒆r + 𝒆θ + 𝒆 𝜕r 𝑟 𝜕θ 𝑟 sin θ 𝜕ϕ ϕ 𝜕𝑓𝑖 𝒆𝒆 𝜕𝑥𝑗 i j where ƒ = (ƒ𝟏 , ƒ𝟐 , ƒ𝟑 ) ∇𝑓 = 𝑏 (Contour Integral if Complex) ∫ 𝑓 𝑑𝑠 = ∫ 𝑓(𝒓(𝑡)) |𝒓′ (𝑡)| 𝑑𝑡 ∫ 𝑓(𝑥, 𝑦) 𝑑𝑠 = Line Integral 𝐶 𝐶 𝑏 𝑑𝑥 2 𝑑𝑦 2 ∫ 𝑓 (𝑥, 𝑦)√( ) + ( ) 𝑑𝑡 𝑑𝑡 𝑑𝑡 𝑎 𝑏 ∫ 𝑭(𝑟) • 𝑑𝑟 = ∫ 𝑭(𝒓(𝑡)) • 𝒓′ (𝑡)𝑑𝑡 = 𝑭(𝑏) − 𝑭(𝑎) 𝐶 𝑎 Fundamental Theorem for Line Integrals 𝑎 ∫ 𝑓 𝑑𝑆 = 𝑆 𝜕𝒙 𝜕𝒙 | 𝑑𝑠 𝑑𝑡 × 𝜕s 𝜕t Where 𝒙(𝑠, 𝑡) = (𝑥 (𝑠, 𝑡), 𝑦(𝑠, 𝑡), 𝑧(𝑠, 𝑡)) and 𝜕𝒙 𝜕𝒙 ( )= × 𝜕s 𝜕t ∫ 𝒗 • 𝑑𝑺 = 𝑆 ∬ 𝑓(𝒙(𝑠, 𝑡)) | 𝑇 Surface Integral 𝜕(𝑦, 𝑧) 𝜕(𝑧, 𝑥) 𝜕(𝑥, 𝑦) ( ) , , 𝜕(𝑠, 𝑡) 𝜕(𝑠, 𝑡) 𝜕(𝑠, 𝑡) Copyright © 2011-2025 by Harold Toomey, WyzAnt Tutor ∫ (𝒗 • 𝒏) 𝑑𝑺 = 𝑺 ∬ 𝒗(𝒙(𝑠, 𝑡)) 𝑇 •( 𝜕𝒙 𝜕𝒙 ) 𝑑𝑠 𝑑𝑡 × 𝜕s 𝜕t 13