Math Calculus Cheat Sheet

PERIMETER, AREA & VOLUME Rectangle 𝑃𝑃 = 2𝑙𝑙 + 2𝑤𝑤 𝐴𝐴 = 𝑙𝑙𝑙𝑙 Square 𝑃𝑃 = 4𝑠𝑠 𝐴𝐴 = 𝑠𝑠 2 Triangle P = add all sides 1 2 A = 𝑏𝑏ℎ Parallelogram P= add all sides A = bh Trapezoid P = add all sides 1 A= (𝑏𝑏1 + 𝑏𝑏2 )h 2 Circle 𝐶𝐶 = 𝜋𝜋𝜋𝜋 = 2𝜋𝜋𝜋𝜋 𝐴𝐴 = 𝜋𝜋𝑟𝑟 2 Arc Length S = 𝜃𝜃𝜃𝜃 in radians 𝜋𝜋 S= 180 𝜃𝜃𝜃𝜃in degrees Circle Sector Area 𝜃𝜃 2 A= 𝜃𝜃 𝜋𝜋𝑟𝑟 2 in degrees 360 A = 𝑟𝑟 2 in radians Rectangular solid S = 2𝑙𝑙𝑙𝑙 + 2𝑙𝑙ℎ + 2𝑤𝑤ℎ V = 𝑙𝑙𝑙𝑙ℎ Cube 𝑆𝑆𝑆𝑆 = 6𝑠𝑠 2 V = s3 Cylinder 𝑆𝑆𝑆𝑆 = 2𝜋𝜋𝑟𝑟 2 + 2𝜋𝜋𝑟𝑟ℎ 𝑉𝑉 = 𝜋𝜋𝑟𝑟 2 ℎ Cone 𝑆𝑆𝑆𝑆 = 𝜋𝜋𝑟𝑟𝑟𝑟 + 𝜋𝜋𝑟𝑟 2 𝑉𝑉 = 1 2 𝜋𝜋𝑟𝑟 ℎ 3 A = 𝜋𝜋𝜋𝜋√𝑟𝑟 2 + ℎ2 Sphere SA = 4πr 2 4 V = πr 3 3 A = 4𝜋𝜋𝜋𝜋 2 Rectangle Pyramid SA = 𝑙𝑙𝑙𝑙 + 2𝑙𝑙𝑙𝑙 + 2𝑤𝑤𝑤𝑤 1 V = 𝑙𝑙𝑙𝑙ℎ 3 PYTHAGOREAN THEOREM leg2+ leg2 = hypotenuse2 DISTANCE FORMULA EXPONENT LAWS 𝑥𝑥 0 = 1 if x ≠ 0 1 𝑥𝑥 = 𝑥𝑥 𝑥𝑥 −𝑛𝑛 𝑚𝑚 = 𝑛𝑛 1 𝑥𝑥 𝑛𝑛 if x ≠ 0 𝑚𝑚+𝑛𝑛 𝑥𝑥 . 𝑥𝑥 = 𝑥𝑥 (𝑥𝑥 𝑚𝑚 )𝑛𝑛 = 𝑥𝑥 𝑚𝑚.𝑛𝑛 𝑥𝑥 𝑚𝑚 ÷ 𝑥𝑥 𝑛𝑛 = 𝑥𝑥 𝑚𝑚 = 𝑥𝑥 𝑚𝑚−𝑛𝑛 if x ≠0 𝑥𝑥 𝑛𝑛 𝑚𝑚 𝑚𝑚 (𝑥𝑥𝑥𝑥)𝑚𝑚 = 𝑥𝑥 𝑦𝑦 𝑥𝑥 𝑛𝑛 𝑦𝑦 𝑑𝑑 = �(𝑥𝑥2 − 𝑥𝑥1 )2 + (𝑦𝑦2 − 𝑦𝑦1 )2 𝑥𝑥 𝑛𝑛 � � = 𝑦𝑦𝑛𝑛 if y ≠0 𝑚𝑚 𝑛𝑛 𝑛𝑛 𝑥𝑥 = √𝑥𝑥 𝑚𝑚 if (a ≥ 0, m ≥0, n>0) PROPERTIES OF LOGARITHMS 𝑦𝑦 = 𝑙𝑙𝑙𝑙𝑙𝑙𝑎𝑎 𝑥𝑥 ⇔ 𝑥𝑥 = 𝑎𝑎 𝑦𝑦 where a >0, a ≠ 0 𝑎𝑎 𝑙𝑙𝑙𝑙𝑙𝑙𝑎𝑎 𝑀𝑀 = 𝑀𝑀 𝑙𝑙𝑙𝑙𝑙𝑙𝒂𝒂 (𝑀𝑀𝑀𝑀) = 𝑙𝑙𝑙𝑙𝑙𝑙𝑎𝑎 𝑀𝑀 + 𝑙𝑙𝑙𝑙𝑙𝑙𝑎𝑎 N 𝑀𝑀 𝑙𝑙𝑙𝑙𝑙𝑙𝒂𝒂 � � = 𝑙𝑙𝑙𝑙𝑙𝑙𝑎𝑎 𝑀𝑀 − 𝑙𝑙𝑙𝑙𝑙𝑙𝑎𝑎 𝑁𝑁 𝑁𝑁 𝑙𝑙𝑙𝑙𝑙𝑙𝑎𝑎 𝑀𝑀 𝑥𝑥 = 𝑥𝑥𝑙𝑙𝑙𝑙𝑙𝑙𝑎𝑎 𝑀𝑀 𝑙𝑙𝑙𝑙𝑙𝑙𝒂𝒂 𝑀𝑀 = 𝑙𝑙𝑙𝑙𝑙𝑙𝑏𝑏 𝑀𝑀 𝑙𝑙𝑙𝑙𝑙𝑙𝑏𝑏 𝒂𝒂 SPECIAL PRODUCTS = 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 𝑙𝑙𝑙𝑙𝑙𝑙𝒂𝒂 = 𝑙𝑙𝑙𝑙𝑙𝑙 𝑙𝑙𝑙𝑙𝒂𝒂 𝑥𝑥 2 − 𝑦𝑦 2 = (𝑥𝑥 + 𝑦𝑦)(𝑥𝑥 − 𝑦𝑦) 𝑥𝑥 3 ± 𝑦𝑦 3 = (𝑥𝑥 ± 𝑦𝑦)(𝑥𝑥 2 ∓ 𝑥𝑥𝑥𝑥 + 𝑦𝑦 2 ) BINOMIAL THEOREM (𝑥𝑥 ± 𝑦𝑦)2 = 𝑥𝑥 2 ± 2𝑥𝑥𝑥𝑥 + 𝑦𝑦 2 (𝑥𝑥 ± 𝑦𝑦)3 = 𝑥𝑥 3 ± 3𝑥𝑥 2 𝑦𝑦 + 3𝑥𝑥𝑦𝑦 2 ± 𝑦𝑦 3 (𝑥𝑥 + 𝑦𝑦)𝑛𝑛 = 𝑥𝑥 𝑛𝑛 + 𝑛𝑛𝑥𝑥 𝑛𝑛−1 𝑦𝑦 𝑛𝑛 𝑛𝑛(𝑛𝑛−1) 𝑛𝑛−2 2 + 𝑥𝑥 𝑦𝑦 + … + � � 𝑥𝑥 𝑛𝑛−𝑘𝑘 2 𝑘𝑘 + … + 𝑛𝑛𝑛𝑛𝑦𝑦 𝑛𝑛−1 + 𝑦𝑦 𝑛𝑛 𝑛𝑛(𝑛𝑛−1)…(𝑛𝑛−𝑘𝑘+1) 𝑛𝑛 where � � = 𝑘𝑘 1•2•3•…•𝑘𝑘 𝑜𝑜𝑜𝑜𝑜𝑜 sin𝛉𝛉 = ℎ𝑦𝑦𝑦𝑦 𝑎𝑎𝑎𝑎𝑎𝑎 𝑐𝑐𝑐𝑐𝑐𝑐𝜽𝜽 = ℎ𝑦𝑦𝑦𝑦 𝑜𝑜𝑜𝑜𝑜𝑜 𝑡𝑡𝑡𝑡𝑡𝑡𝜽𝜽 = 𝑎𝑎𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎𝑎𝑎 𝑐𝑐𝑐𝑐𝑐𝑐𝜽𝜽 = 𝑜𝑜𝑜𝑜𝑜𝑜 ℎ𝑦𝑦𝑦𝑦 𝑐𝑐𝑐𝑐𝑐𝑐𝜽𝜽 = 𝑜𝑜𝑜𝑜𝑜𝑜 ℎ𝑦𝑦𝑦𝑦 𝑠𝑠𝑠𝑠𝑠𝑠𝜽𝜽 = 𝑎𝑎𝑎𝑎𝑎𝑎 𝑠𝑠𝑠𝑠𝑠𝑠𝜽𝜽 𝑡𝑡𝑡𝑡𝑡𝑡𝜽𝜽 = 𝑐𝑐𝑐𝑐𝑐𝑐𝜽𝜽 𝑐𝑐𝑐𝑐𝑐𝑐𝜽𝜽 𝑐𝑐𝑐𝑐𝑐𝑐𝜽𝜽 = 𝑠𝑠𝑠𝑠𝑠𝑠𝜽𝜽 1 𝑠𝑠𝑠𝑠𝑠𝑠𝜽𝜽 = 𝑐𝑐𝑐𝑐𝑐𝑐𝜽𝜽 1 𝑐𝑐𝑐𝑐𝑐𝑐𝜽𝜽 = 𝑠𝑠𝑠𝑠𝑠𝑠𝜽𝜽 1 𝑐𝑐𝑐𝑐𝑐𝑐𝜽𝜽 = 𝑠𝑠𝑠𝑠𝑠𝑠𝜽𝜽 1 𝑠𝑠𝑠𝑠𝑠𝑠𝜽𝜽 = 𝑐𝑐𝑐𝑐𝑐𝑐𝜽𝜽 1 𝑡𝑡𝑡𝑡𝑡𝑡𝜽𝜽 = 𝑐𝑐𝑐𝑐𝑐𝑐𝜽𝜽 1 𝑐𝑐𝑐𝑐𝑐𝑐𝜽𝜽 = 𝑡𝑡𝑡𝑡𝑡𝑡𝜽𝜽 𝑠𝑠𝑠𝑠𝑠𝑠2 𝜽𝜽 + 𝑐𝑐𝑐𝑐𝑐𝑐 2 𝜽𝜽 = 1 𝑡𝑡𝑡𝑡𝑡𝑡2 𝜽𝜽 + 1 = 𝑠𝑠𝑠𝑠𝑠𝑠 2 𝜽𝜽 𝑐𝑐𝑐𝑐𝑐𝑐 2 𝜽𝜽 + 1 = 𝑐𝑐𝑐𝑐𝑐𝑐 2 𝜽𝜽 1 − 𝑐𝑐𝑐𝑐𝑐𝑐𝟐𝟐𝟐𝟐 𝑠𝑠𝑠𝑠𝑠𝑠2 𝜽𝜽 = 2 1 + 𝑐𝑐𝑐𝑐𝑐𝑐𝟐𝟐𝟐𝟐 2 𝑐𝑐𝑐𝑐𝑐𝑐 𝜽𝜽 = 2 1 − 𝑐𝑐𝑐𝑐𝑐𝑐𝟐𝟐𝟐𝟐 2 𝑡𝑡𝑡𝑡𝑡𝑡 𝜽𝜽 = 1 + 𝑐𝑐𝑐𝑐𝑐𝑐𝟐𝟐𝟐𝟐 𝑠𝑠𝑠𝑠𝑠𝑠(𝟐𝟐𝟐𝟐) = 2𝑠𝑠𝑠𝑠𝑠𝑠𝜽𝜽𝑐𝑐𝑐𝑐𝑐𝑐𝜽𝜽 𝑐𝑐𝑐𝑐𝑐𝑐(𝟐𝟐𝟐𝟐) = 𝑐𝑐𝑐𝑐𝑐𝑐 2 𝜽𝜽 − 𝑠𝑠𝑠𝑠𝑠𝑠2 𝜽𝜽 = 2𝑐𝑐𝑐𝑐𝑐𝑐 2 𝜽𝜽 − 1 = 1 − 2𝑠𝑠𝑠𝑠𝑠𝑠2 𝜽𝜽 𝑡𝑡𝑡𝑡𝑡𝑡(𝟐𝟐𝟐𝟐) = 2𝑡𝑡𝑡𝑡𝑡𝑡𝜽𝜽 1 − 𝑡𝑡𝑡𝑡𝑡𝑡2 𝜽𝜽 Law of Sines 𝑠𝑠𝑠𝑠𝑠𝑠𝜶𝜶 𝑠𝑠𝑠𝑠𝑠𝑠𝜷𝜷 𝑠𝑠𝑠𝑠𝑠𝑠𝜸𝜸 = = 𝑎𝑎 𝑏𝑏 𝑐𝑐 Law of Cosines 𝑎𝑎2 = 𝑏𝑏 2 + 𝑐𝑐 2 − 2𝑏𝑏𝑏𝑏𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄 𝑏𝑏 2 = 𝑎𝑎2 + 𝑐𝑐 2 − 2𝑎𝑎𝑎𝑎𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄 𝑐𝑐 2 = 𝑎𝑎2 + 𝑏𝑏 2 − 2𝑎𝑎𝑎𝑎𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄 Sum and Difference 𝑠𝑠𝑠𝑠𝑠𝑠(𝜶𝜶 ± 𝜷𝜷) = 𝑠𝑠𝑠𝑠𝑠𝑠𝜶𝜶𝑐𝑐𝑐𝑐𝑐𝑐𝜷𝜷 ± 𝑐𝑐𝑐𝑐𝑐𝑐𝜶𝜶𝑠𝑠𝑠𝑠𝑠𝑠𝜷𝜷 𝑐𝑐𝑐𝑐𝑐𝑐(𝜶𝜶 ± 𝜷𝜷) = 𝑐𝑐𝑐𝑐𝑐𝑐𝜶𝜶𝑐𝑐𝑐𝑐𝑐𝑐𝜷𝜷 ∓ 𝑠𝑠𝑠𝑠𝑠𝑠𝜶𝜶𝑠𝑠𝑠𝑠𝑠𝑠𝜷𝜷 Sum to Product 𝜶𝜶 + 𝜷𝜷 𝜶𝜶 − 𝜷𝜷 𝑠𝑠𝑠𝑠𝑠𝑠𝜶𝜶 + 𝑠𝑠𝑠𝑠𝑠𝑠𝜷𝜷 = 2𝑠𝑠𝑠𝑠𝑠𝑠 � � 𝑐𝑐𝑐𝑐𝑐𝑐 � � 𝟐𝟐 𝟐𝟐 𝜶𝜶 − 𝜷𝜷 𝜶𝜶 + 𝜷𝜷 � 𝑠𝑠𝑠𝑠𝑠𝑠 � � 𝑠𝑠𝑠𝑠𝑠𝑠𝜶𝜶 − 𝑠𝑠𝑠𝑠𝑠𝑠𝜷𝜷 = 2𝑐𝑐𝑐𝑐𝑐𝑐 � 𝟐𝟐 𝟐𝟐 𝜶𝜶 − 𝜷𝜷 𝜶𝜶 + 𝜷𝜷 � 𝑐𝑐𝑐𝑐𝑐𝑐 � � 𝑐𝑐𝑐𝑐𝑐𝑐𝜶𝜶 + 𝑐𝑐𝑐𝑐𝑐𝑐𝜷𝜷 = 2𝑐𝑐𝑐𝑐𝑐𝑐 � 𝟐𝟐 𝟐𝟐 𝜶𝜶 + 𝜷𝜷 𝜶𝜶 − 𝜷𝜷 𝑐𝑐𝑐𝑐𝑐𝑐𝜶𝜶 − 𝑐𝑐𝑐𝑐𝑐𝑐𝜷𝜷 = −2𝑠𝑠𝑠𝑠𝑠𝑠 � � 𝑠𝑠𝑠𝑠𝑠𝑠 � � 𝟐𝟐 𝟐𝟐 𝑡𝑡𝑡𝑡𝑡𝑡(𝜶𝜶 ± 𝜷𝜷) = 𝑡𝑡𝑡𝑡𝑡𝑡𝜶𝜶 ± 𝑡𝑡𝑡𝑡𝑡𝑡𝜷𝜷 1 ∓ 𝑡𝑡𝑡𝑡𝑡𝑡𝜶𝜶𝑡𝑡𝑡𝑡𝑡𝑡𝜷𝜷 Product to Sum 𝑐𝑐𝑐𝑐𝑐𝑐(𝜶𝜶 − 𝜷𝜷) − 𝑐𝑐𝑐𝑐𝑐𝑐(𝜶𝜶 + 𝜷𝜷) 2 𝑐𝑐𝑐𝑐𝑐𝑐(𝜶𝜶 − 𝜷𝜷) + 𝑐𝑐𝑐𝑐𝑐𝑐(𝜶𝜶 + 𝜷𝜷) 𝑐𝑐𝑐𝑐𝑐𝑐𝜶𝜶𝑐𝑐𝑐𝑐𝑐𝑐𝜷𝜷 = 2 𝑠𝑠𝑠𝑠𝑠𝑠(𝜶𝜶 + 𝜷𝜷) + 𝑠𝑠𝑠𝑠𝑠𝑠(𝜶𝜶 − 𝜷𝜷) 𝑠𝑠𝑠𝑠𝑠𝑠𝜶𝜶𝑐𝑐𝑐𝑐𝑐𝑐𝜷𝜷 = 2 𝑠𝑠𝑠𝑠𝑠𝑠(𝜶𝜶 + 𝜷𝜷) − 𝑠𝑠𝑠𝑠𝑠𝑠(𝜶𝜶 − 𝜷𝜷) 𝑐𝑐𝑐𝑐𝑐𝑐𝜶𝜶𝑠𝑠𝑠𝑠𝑠𝑠𝜷𝜷 = 2 𝑠𝑠𝑠𝑠𝑠𝑠𝜶𝜶𝑠𝑠𝑠𝑠𝑠𝑠𝜷𝜷 = DERIVATIVES Definition: Derivative: 𝑓𝑓 ′ (𝑥𝑥) COMMON DERIVATIVES 𝑓𝑓(𝑥𝑥+ℎ)−𝑓𝑓(𝑥𝑥) = 𝑙𝑙𝑙𝑙𝑙𝑙 if this limit exists. ℎ→0 ℎ Applications: If 𝑦𝑦 = 𝑓𝑓(𝑥𝑥) then, • 𝑚𝑚 = 𝑓𝑓 ′(𝑎𝑎) is the slope of the tangent line to y=f(x) at x=a and the equation of the tangent line at 𝑥𝑥 = 𝑎𝑎 is given by 𝑦𝑦 = 𝑓𝑓(𝑎𝑎) + 𝑓𝑓 ′(𝑎𝑎)(𝑥𝑥 − 𝑎𝑎). • 𝑓𝑓′(𝑎𝑎)is the instantaneous rate of change of 𝑓𝑓(𝑥𝑥)at 𝑥𝑥 = 𝑎𝑎. • If 𝑓𝑓(𝑥𝑥)is the position of an object at time 𝑥𝑥, then 𝑓𝑓 ′(𝑎𝑎) is the velocity of the object at 𝑥𝑥 = 𝑎𝑎 Critical points: 𝑥𝑥 = 𝑐𝑐 is the critical point of 𝑓𝑓(𝑥𝑥) = 𝑐𝑐 provided either 1. 𝑓𝑓 ′(𝑐𝑐) = 0 or 2. 𝑓𝑓 ′(𝑐𝑐) does not exist. Increasing/Decreasing • If 𝑓𝑓 ′(𝑥𝑥) > 0 for all x in an interval I, then f(x) is increasing on the interval I. • If 𝑓𝑓 ′(𝑥𝑥) < 0 for all x in an interval I, then f(x) is decreasing on the interval I. • If 𝑓𝑓 ′ (𝑥𝑥) = 0 for all x in an interval I, then f(x) is constant on the interval I. Concavity • If 𝑓𝑓 ′′(𝑥𝑥) > 0 for all x in an interval I, then f(x) is concave up on the interval I. • If 𝑓𝑓 ′′(𝑥𝑥) < 0 for all x in an interval I, then f(x) is concave down on the interval I. Inflection Points 𝑥𝑥 = 𝑐𝑐 is an inflection point of f(x)if the concavity changes at 𝑥𝑥 = 𝑐𝑐. INTEGRATION 1) 𝑐𝑐’ = 0 2) [𝑓𝑓(𝑥𝑥) + 𝑔𝑔(𝑥𝑥)]’ = 𝑓𝑓’(𝑥𝑥) + 𝑔𝑔’(𝑥𝑥) 3) [𝑓𝑓(𝑥𝑥)𝑔𝑔(𝑥𝑥)]’ = 𝑓𝑓(𝑥𝑥)𝑔𝑔’(𝑥𝑥) + 𝑓𝑓’(𝑥𝑥)𝑔𝑔(𝑥𝑥) 4) [𝑓𝑓(𝑔𝑔(𝑥𝑥))]’ = 𝑓𝑓’(𝑔𝑔(𝑥𝑥))𝑔𝑔’(𝑥𝑥) 5) [𝑐𝑐𝑐𝑐(𝑥𝑥)]’ = 𝑐𝑐𝑐𝑐’(𝑥𝑥) 6) [𝑓𝑓(𝑥𝑥) – 𝑔𝑔(𝑥𝑥)]’ = 𝑓𝑓’(𝑥𝑥) – 𝑔𝑔’(𝑥𝑥) 𝑓𝑓(𝑥𝑥) ′ 𝑔𝑔(𝑥𝑥)𝑓𝑓′(𝑥𝑥)− 𝑓𝑓(𝑥𝑥)𝑔𝑔′ (𝑥𝑥) 7) � = � 𝑔𝑔(𝑥𝑥) [𝑔𝑔(𝑥𝑥)]2 𝑛𝑛 𝑛𝑛−1 (𝑥𝑥 )′ = 𝑛𝑛𝑥𝑥 [𝑒𝑒 𝑥𝑥 ]′ = 𝑒𝑒 𝑥𝑥 10) [𝑎𝑎 𝑥𝑥 ]′ = 𝑎𝑎 𝑥𝑥 𝑙𝑙𝑙𝑙𝒂𝒂 8) 9) 11) [𝑙𝑙𝑙𝑙|𝒙𝒙|]′ = 12) [𝑙𝑙𝑙𝑙𝑙𝑙𝑎𝑎 𝒙𝒙]′ Definition: Suppose 𝑓𝑓(𝑥𝑥) is continuous on [𝑎𝑎, 𝑏𝑏]. Divide [𝑎𝑎, 𝑏𝑏] into n subintervals of width ∆𝑥𝑥 and choose 𝑥𝑥𝑖𝑖∗ from each interval. Then 𝑏𝑏 ∞ � 𝑓𝑓(𝑥𝑥)𝑑𝑑𝑑𝑑 = lim � 𝑓𝑓(𝑥𝑥𝑖𝑖∗ )∆𝑥𝑥 𝑎𝑎 𝑛𝑛→∞ 𝑖𝑖=1 where ∆𝑥𝑥 Fundamental Theorem of Calculus: Suppose 𝑓𝑓(𝑥𝑥) is continuous on [𝑎𝑎, 𝑏𝑏], then 𝑥𝑥 Part I: 𝑔𝑔(𝑥𝑥) =∫𝑎𝑎 𝑓𝑓(𝑡𝑡)𝑑𝑑𝑑𝑑 is also continuous on [𝑎𝑎, 𝑏𝑏] and 𝑔𝑔′ (𝑥𝑥) = 𝑑𝑑 𝑏𝑏 𝑑𝑑 𝑥𝑥 = (𝑏𝑏−𝑎𝑎) 𝑛𝑛 ∫ 𝑓𝑓(𝑡𝑡)𝑑𝑑𝑑𝑑 = 𝑓𝑓(𝑥𝑥)where 𝑎𝑎 ≤ 𝑥𝑥 ≤ 𝑏𝑏. 𝑑𝑑𝑑𝑑 𝑎𝑎 Part II: 𝑑𝑑𝑑𝑑 ∫𝑎𝑎 𝑓𝑓(𝑥𝑥)𝑑𝑑𝑑𝑑 = 𝐹𝐹(𝑏𝑏) − 𝐹𝐹(𝑎𝑎) where 𝐹𝐹(𝑥𝑥) is any anti-derivative of 𝑓𝑓(𝑥𝑥), i.e, a function such that 𝐹𝐹’ = 𝑓𝑓. Applications: 𝑏𝑏 Area: 𝐴𝐴 = ∫𝑎𝑎 𝑓𝑓(𝑥𝑥)𝑑𝑑𝑑𝑑 Area between Curves: 𝑏𝑏 1 INTEGRALS 1) ∫ 𝑢𝑢𝑛𝑛 𝑑𝑑𝑑𝑑 = 2) ∫ 𝑑𝑑𝑑𝑑 𝑢𝑢 19) (𝑠𝑠𝑠𝑠𝑠𝑠−1 𝒙𝒙)′ = 22) (𝑐𝑐𝑐𝑐𝑐𝑐 −1 𝒙𝒙) ′ = − 1 1+𝑥𝑥 2 21) (𝑡𝑡𝑡𝑡𝑡𝑡−1 𝒙𝒙)′ = 18) + 𝑐𝑐, 𝑛𝑛 ≠ −1 𝑏𝑏 Cylinders V= ∫𝑎𝑎 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 ∙ ℎ𝑒𝑒𝑒𝑒𝑒𝑒ℎ𝑡𝑡 ∙ 𝑡𝑡ℎ𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑏𝑏 Work: If a force of 𝐹𝐹(𝑥𝑥) moves an object in 𝑎𝑎 ≤ 𝑥𝑥 ≤ 𝑏𝑏, then the work done is W=∫𝑎𝑎 𝐹𝐹(𝑥𝑥)𝑑𝑑𝑑𝑑 𝑏𝑏 ∫ 𝑓𝑓(𝑥𝑥)𝑑𝑑𝑑𝑑 𝑏𝑏−𝑎𝑎 𝑎𝑎 1 34) (𝑐𝑐𝑐𝑐𝑐𝑐ℎ−1 𝒙𝒙)′ = 1 1−𝑥𝑥 2 33) (𝑡𝑡𝑡𝑡𝑡𝑡ℎ−1 𝒙𝒙)′ = 1 35) (𝑠𝑠𝑠𝑠𝑠𝑠ℎ−1 𝒙𝒙)′ = − |𝑥𝑥|√𝑥𝑥 2 −1 1 1 1−𝑥𝑥 2 36) (𝑐𝑐𝑐𝑐𝑐𝑐ℎ−1 𝒙𝒙)′ = − |𝑥𝑥|�𝑥𝑥2 −1 𝑑𝑑𝑑𝑑 1 𝒖𝒖 ∫ 𝑎𝑎2 +𝑢𝑢2 = 𝑎𝑎 𝑡𝑡𝑡𝑡𝑡𝑡−1 𝒂𝒂 + 𝑐𝑐 1 |𝑥𝑥|√1−𝑥𝑥 2 1 |𝑥𝑥|�𝑥𝑥2 +1 𝑑𝑑𝑑𝑑 𝒖𝒖 1 ∫ 𝑢𝑢√𝑢𝑢2 −𝑎𝑎2 = 𝑎𝑎 𝑠𝑠𝑠𝑠𝑠𝑠 −1 𝒂𝒂 + 𝑐𝑐 𝑑𝑑𝑑𝑑 1 𝒖𝒖+𝒂𝒂 1 𝒖𝒖−𝒂𝒂 ∫ 𝒂𝒂𝟐𝟐−𝒖𝒖𝟐𝟐 = 2𝑎𝑎 𝑙𝑙𝑙𝑙 �𝒖𝒖−𝒂𝒂�+c ∫ 𝑙𝑙𝑙𝑙𝒖𝒖 𝑑𝑑𝑑𝑑 = 𝑢𝑢𝑢𝑢𝑢𝑢𝒖𝒖 − 𝑢𝑢 + 𝑐𝑐 22) ∫ 𝑠𝑠𝑠𝑠𝑠𝑠−1 𝒖𝒖 𝑑𝑑𝑑𝑑 = 𝑢𝑢 𝑠𝑠𝑠𝑠𝑠𝑠−1 𝒖𝒖 + √1 − 𝑢𝑢 2 + 𝑐𝑐 𝑎𝑎 𝑢𝑢 +c 𝑙𝑙𝑙𝑙𝒂𝒂 1 6) ∫ 𝑑𝑑𝑑𝑑 𝑢𝑢𝑢𝑢𝑢𝑢𝒖𝒖 = 𝑙𝑙𝑙𝑙|𝑙𝑙𝑙𝑙𝒖𝒖| + 𝑐𝑐 13) ∫ 𝑠𝑠𝑠𝑠𝑠𝑠 2 𝒖𝒖 𝑑𝑑𝑑𝑑 = 𝑡𝑡𝑡𝑡𝑡𝑡 𝒖𝒖 + 𝑐𝑐 14) ∫ 𝑐𝑐𝑐𝑐𝑐𝑐 2 𝒖𝒖 𝑑𝑑𝑑𝑑 = −𝑐𝑐𝑐𝑐𝑐𝑐𝒖𝒖 + 𝑐𝑐 15) ∫ 𝑠𝑠𝑠𝑠𝑠𝑠𝒖𝒖 𝑡𝑡𝑡𝑡𝑡𝑡𝒖𝒖 𝑑𝑑𝑑𝑑 = 𝑠𝑠𝑠𝑠𝑠𝑠𝒖𝒖 + 𝑐𝑐 16) ∫ 𝑐𝑐𝑐𝑐𝑐𝑐𝒖𝒖 𝑐𝑐𝑐𝑐𝑐𝑐𝒖𝒖 𝑑𝑑𝑑𝑑 = −𝑐𝑐𝑐𝑐𝑐𝑐𝒖𝒖 + 𝑐𝑐 17) 19) √𝑥𝑥 2 −1 20) 12) ∫ 𝑐𝑐𝑐𝑐𝑐𝑐𝒖𝒖 𝑑𝑑𝑑𝑑 = 𝑙𝑙𝑙𝑙|𝑐𝑐𝑐𝑐𝑐𝑐𝒖𝒖 − 𝑐𝑐𝑐𝑐𝑐𝑐𝒖𝒖| + 𝑐𝑐 Rings V= ∫𝑎𝑎 2𝜋𝜋(𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜 𝑟𝑟 2 − 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑟𝑟 2 ) 1 1+𝑥𝑥 2 1 32) (𝑐𝑐𝑐𝑐𝑐𝑐ℎ−1 𝒙𝒙)′ = ∫ 𝑒𝑒 𝑢𝑢 𝑑𝑑𝑑𝑑 = 𝑒𝑒 𝑢𝑢 + 𝑐𝑐 11) ∫ 𝑠𝑠𝑠𝑠𝑠𝑠𝒖𝒖 𝑑𝑑𝑑𝑑 = 𝑙𝑙𝑙𝑙|𝑠𝑠𝑠𝑠𝑠𝑠𝒖𝒖 + 𝑡𝑡𝑡𝑡𝑡𝑡𝒖𝒖| + 𝑐𝑐 Volume of Revolution 1 √1−𝑥𝑥 2 10) ∫ 𝑐𝑐𝑐𝑐𝑐𝑐𝒖𝒖 𝑑𝑑𝑑𝑑 = 𝑙𝑙𝑙𝑙|𝑠𝑠𝑠𝑠𝑠𝑠𝒖𝒖| + 𝑐𝑐 Volumes: V= ∫𝑎𝑎 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴(𝑥𝑥) 𝑑𝑑𝑑𝑑 1 √1−𝑥𝑥 2 20) (𝑐𝑐𝑐𝑐𝑐𝑐 −1 𝒙𝒙) ′ = − = 𝑙𝑙𝑙𝑙|𝒖𝒖| + 𝑐𝑐 4) ∫ 𝑎𝑎 𝑢𝑢 𝑑𝑑𝑑𝑑 = 5) 𝑛𝑛+1 28) (𝑐𝑐𝑐𝑐𝑐𝑐ℎ𝒙𝒙)′ = −𝑐𝑐𝑐𝑐𝑐𝑐ℎ2 𝒙𝒙 29) (𝑠𝑠𝑠𝑠𝑠𝑠ℎ𝒙𝒙)′ = −𝑠𝑠𝑠𝑠𝑠𝑠ℎ𝒙𝒙𝑡𝑡𝑡𝑡𝑡𝑡ℎ𝒙𝒙 30) (𝑐𝑐𝑐𝑐𝑐𝑐𝒙𝒙)′ = −𝑐𝑐𝑐𝑐𝑐𝑐ℎ𝒙𝒙𝑐𝑐𝑐𝑐𝑐𝑐ℎ𝒙𝒙 1 31) (𝑠𝑠𝑠𝑠𝑠𝑠ℎ−1 𝒙𝒙)′ = √1+𝑥𝑥 2 = −𝑐𝑐𝑐𝑐𝑐𝑐 𝒙𝒙 17) = 𝑠𝑠𝑠𝑠𝑠𝑠𝒙𝒙𝑡𝑡𝑡𝑡𝑡𝑡𝒙𝒙 18) (𝑐𝑐𝑐𝑐𝑐𝑐𝒙𝒙)′ = −𝑐𝑐𝑐𝑐𝑐𝑐𝒙𝒙𝑐𝑐𝑐𝑐𝑐𝑐𝒙𝒙 (𝑠𝑠𝑠𝑠𝑠𝑠𝒙𝒙)′ 24) (𝑐𝑐𝑐𝑐𝑐𝑐 −1 𝒙𝒙)′ = − 𝑢𝑢𝑛𝑛+1 27) (𝑡𝑡𝑡𝑡𝑡𝑡ℎ𝒙𝒙) ′ = 𝑠𝑠𝑠𝑠𝑠𝑠ℎ2 𝒙𝒙 2 23) (𝑠𝑠𝑠𝑠𝑠𝑠 −1 𝒙𝒙)′ = 𝑥𝑥𝑥𝑥𝑥𝑥𝒂𝒂 3) 16) (𝑐𝑐𝑐𝑐𝑐𝑐𝒙𝒙)′ 9) ∫ 𝑡𝑡𝑡𝑡𝑡𝑡𝒖𝒖 𝑑𝑑𝑑𝑑 = 𝑙𝑙𝑙𝑙|𝑠𝑠𝑠𝑠𝑠𝑠𝒖𝒖| + 𝑐𝑐 𝑏𝑏 Average Function Value: The average value of 𝑓𝑓(𝑥𝑥) on 𝑎𝑎 ≤ 𝑥𝑥 ≤ 𝑏𝑏 is𝑓𝑓𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 = = 26) (𝑐𝑐𝑐𝑐𝑐𝑐ℎ𝒙𝒙)′ = 𝑠𝑠𝑠𝑠𝑠𝑠ℎ𝒙𝒙 15) (𝑡𝑡𝑡𝑡𝑡𝑡𝒙𝒙) ′ = 𝑠𝑠𝑠𝑠𝑠𝑠 2 𝒙𝒙 8) ∫ 𝑐𝑐𝑐𝑐𝑐𝑐𝒖𝒖 𝑑𝑑𝑑𝑑 = 𝑠𝑠𝑠𝑠𝑠𝑠𝒖𝒖 + 𝑐𝑐 • 𝑥𝑥 = 𝑓𝑓(𝑦𝑦); 𝐴𝐴 = ∫𝑎𝑎 (𝑟𝑟𝑟𝑟𝑟𝑟ℎ𝑡𝑡 − 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓) 𝑑𝑑𝑑𝑑 𝑏𝑏 𝑥𝑥 25) (𝑠𝑠𝑠𝑠𝑠𝑠ℎ𝒙𝒙)′ = cosh𝒙𝒙 14) (𝑐𝑐𝑐𝑐𝑐𝑐𝒙𝒙)′ = −𝑠𝑠𝑠𝑠𝑠𝑠𝒙𝒙 7) ∫ 𝑠𝑠𝑠𝑠𝑠𝑠𝒖𝒖 𝑑𝑑𝑑𝑑 = −𝑐𝑐𝑐𝑐𝑐𝑐𝒖𝒖 + 𝑐𝑐 • 𝑦𝑦 = 𝑓𝑓(𝑥𝑥); 𝐴𝐴 = ∫𝑎𝑎 (𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢 − 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓)𝑑𝑑𝑑𝑑 𝑏𝑏 1 13) (𝑠𝑠𝑠𝑠𝑠𝑠𝒙𝒙)′ = cos𝒙𝒙 𝑑𝑑𝑑𝑑 𝒖𝒖 ∫ √𝑎𝑎2 −𝑢𝑢2 = 𝑠𝑠𝑠𝑠𝑠𝑠−1 𝒂𝒂 + 𝑐𝑐, 𝑎𝑎 > 0 21) 𝑑𝑑𝑑𝑑 ∫ 𝒖𝒖𝟐𝟐−𝒂𝒂𝟐𝟐 = 2𝑎𝑎 𝑙𝑙𝑙𝑙 �𝒖𝒖+𝒂𝒂�+c 23) ∫ 𝑐𝑐𝑐𝑐𝑐𝑐 −1 𝒖𝒖 𝑑𝑑𝑑𝑑 = 𝑢𝑢 𝑐𝑐𝑐𝑐𝑐𝑐 −1 𝒖𝒖 + √1 − 𝑢𝑢 2 + 𝑐𝑐 1 24) ∫ 𝑡𝑡𝑡𝑡𝑡𝑡−1 𝒖𝒖 𝑑𝑑𝑑𝑑 = 𝑢𝑢 tan−1 𝒖𝒖 − ln(1 + 𝑢𝑢2 ) + 𝑐𝑐 25) ∫ 𝑠𝑠𝑠𝑠𝑠𝑠ℎ𝒖𝒖 𝑑𝑑𝑑𝑑 = 𝑐𝑐𝑐𝑐𝑐𝑐ℎ𝒖𝒖 + 𝑐𝑐 26) ∫ 𝑐𝑐𝑐𝑐𝑐𝑐ℎ𝒖𝒖 𝑑𝑑𝑑𝑑 = 𝑠𝑠𝑠𝑠𝑠𝑠ℎ𝒖𝒖 + 𝑐𝑐 2 27) ∫ 𝑡𝑡𝑡𝑡𝑡𝑡ℎ𝒖𝒖 𝑑𝑑𝑑𝑑 = 𝑙𝑙𝑙𝑙(𝑐𝑐𝑐𝑐𝑐𝑐ℎ𝒖𝒖) + 𝑐𝑐 28) ∫ 𝑐𝑐𝑐𝑐𝑐𝑐ℎ𝒖𝒖 𝑑𝑑𝑑𝑑 = 𝑙𝑙𝑙𝑙|𝑠𝑠𝑠𝑠𝑠𝑠ℎ𝒖𝒖| + 𝑐𝑐 29) ∫ 𝑠𝑠𝑠𝑠𝑠𝑠ℎ𝒖𝒖 𝑑𝑑𝑑𝑑 = 𝑡𝑡𝑡𝑡𝑡𝑡−1 |𝑠𝑠𝑠𝑠𝑠𝑠ℎ𝒖𝒖| + 𝑐𝑐 1 30) ∫ 𝑐𝑐𝑐𝑐𝑐𝑐ℎ𝒖𝒖 𝑑𝑑𝑑𝑑 = 𝑙𝑙𝑙𝑙 �𝑡𝑡𝑡𝑡𝑡𝑡ℎ 𝒖𝒖� + 𝑐𝑐 2 31) ∫ 𝑠𝑠𝑠𝑠𝑠𝑠ℎ2 𝒖𝒖 𝑑𝑑𝑑𝑑 = 𝑡𝑡𝑡𝑡𝑡𝑡ℎ 𝒖𝒖 + 𝑐𝑐 32) ∫ 𝑐𝑐𝑐𝑐𝑐𝑐ℎ2 𝒖𝒖 𝑑𝑑𝑑𝑑 = −𝑐𝑐𝑐𝑐𝑐𝑐ℎ𝒖𝒖 + 𝑐𝑐 33) ∫ 𝑠𝑠𝑠𝑠𝑠𝑠ℎ𝒖𝒖 𝑡𝑡𝑡𝑡𝑡𝑡ℎ𝒖𝒖 𝑑𝑑𝑑𝑑 = −𝑠𝑠𝑠𝑠𝑠𝑠ℎ𝒖𝒖 + 𝑐𝑐 34) ∫ 𝑐𝑐𝑐𝑐𝑐𝑐ℎ𝒖𝒖 𝑐𝑐𝑐𝑐𝑐𝑐ℎ𝒖𝒖 𝑑𝑑𝑑𝑑 = −𝑐𝑐𝑐𝑐𝑐𝑐ℎ𝒖𝒖 + 𝑐𝑐 35) ∫ 𝑢𝑢𝑢𝑢𝑢𝑢 = 𝑢𝑢𝑢𝑢 − ∫ 𝑣𝑣𝑣𝑣𝑣𝑣