Calculus Cheat Sheet.pdf

7 13 4 4 3 16 a b c b a c a b c an alternate method

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7 13 4 0 4 3 16 A B C B A C A B C + = = = = = = An alternate method that sometimes works to find constants. Start with setting numerators equal in previous example : ( ) ( ) ( ) 2 2 7 13 4 1 x x A x Bx C x + = + + + . Chose nice values of x and plug in. For example if 1 x = we get 20 5 A = which gives 4 A = . This won’t always work easily.
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Calculus Cheat Sheet Visit http://tutorial.math.lamar.edu for a complete set of Calculus notes. © 2005 Paul Dawkins Applications of Integrals Net Area : ( ) b a f x dx represents the net area between ( ) f x and the x -axis with area above x -axis positive and area below x -axis negative. Area Between Curves : The general formulas for the two main cases for each are, ( ) upper function lower function b a y f x A dx = = & ( ) right function left function d c x f y A dy = = If the curves intersect then the area of each portion must be found individually. Here are some sketches of a couple possible situations and formulas for a couple of possible cases. ( ) ( ) b a A f x g x dx = ( ) ( ) d c A f y g y dy = ( ) ( ) ( ) ( ) c b a c A f x g x dx g x f x dx = + Volumes of Revolution : The two main formulas are ( ) V A x dx = and ( ) V A y dy = . Here is some general information about each method of computing and some examples. Rings Cylinders ( ) ( ) ( ) 2 2 outer radius inner radius A π = ( ) ( ) radius width / height 2 A π = Limits: x / y of right/bot ring to x / y of left/top ring Limits : x / y of inner cyl. to x / y of outer cyl. Horz. Axis use ( ) f x , ( ) g x , ( ) A x and dx . Vert. Axis use ( ) f y , ( ) g y , ( ) A y and dy . Horz. Axis use ( ) f y , ( ) g y , ( ) A y and dy . Vert. Axis use ( ) f x , ( ) g x , ( ) A x and dx . Ex. Axis : 0 y a = > Ex. Axis : 0 y a = Ex. Axis : 0 y a = > Ex. Axis : 0 y a = outer radius : ( ) a f x inner radius : ( ) a g x outer radius: ( ) a g x + inner radius: ( ) a f x + radius : a y width : ( ) ( ) f y g y radius : a y + width : ( ) ( ) f y g y These are only a few cases for horizontal axis of rotation. If axis of rotation is the x -axis use the 0 y a = case with 0 a = . For vertical axis of rotation ( 0 x a = > and 0 x a = ) interchange x and y to get appropriate formulas.
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Calculus Cheat Sheet Visit http://tutorial.math.lamar.edu for a complete set of Calculus notes. © 2005 Paul Dawkins Work : If a force of ( ) F x moves an object in a x b , the work done is ( ) b a W F x dx = Average Function Value : The average value of ( ) f x on a x b is ( ) 1 b avg a b a f f x dx = Arc Length Surface Area : Note that this is often a Calc II topic. The three basic formulas are, b a L ds = 2 b a SA y ds π = (rotate about x -axis) 2 b a SA x ds π = (rotate about y -axis) where ds is dependent upon the form of the function being worked with as follows. ( ) ( ) 2 1 if , dy dx ds dx y f x a x b = + = ( ) ( ) 2 1 if , dx dy ds dy x f y a y b = + = ( ) ( ) ( ) ( ) 2 2 if , , dy dx dt dt ds dt x f t y g t a t b = + = = ( ) ( ) 2 2 if , dr d ds r d r f a b θ θ θ θ = + = With surface area you may have to substitute in for the x or y depending on your choice of ds to match the differential in the ds . With parametric and polar you will always need to substitute.
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