Natural Log Rules lne 1 ln1 0 ln0 undefined ln 1 a lna ln a b lna lnb ln ab lna lnb Chapter 7 Integration Fundamental Theorem of Calculus Part 1 If F is an antiderivative of f F f Then Part 2 If f is continuous Then is an antiderivative of f 7 1 Integration by Substitution 1 Let w be the inside function 2 Reunite the integral as an integral in w 3 Evaluate the integral in w 4 convert w back to x 7 2 Integration by Parts Choices for u in order of preference Logarithm Inverse Trig Polynomial Exponential Trig 7 3 Tables of Integrals Completing the Square Trig Functions sin 2x 2 sinx cosx cos 2x cos2x sin2x 7 4 Algebraic Identities and Trigonometric Substitutions Partial Fractions Strategy for Integrating Rational Function P x Q x 1 If deg deg Q use algebraic long division and use partial fractions on the remainder 2 If Q x is a product of distinct linear factors use partial fractions of the form 3 IF Q x has a repeated linear factor x c n use partial fractions of the form 4 If Q x contains an unfactorable quadratic factor q x try a partial fraction of the form Trigonometric Substitutions Sine Substitutions Use to simplify integrands involving Let x asin Use to simplify integrands involving a2 x2 Let x atan Tangent Substitutions 7 5 Approximating Definite Integrals Midpoint Rule Trapezoidal Rule Under or Over Estimate If f is increasing on a b then If f is decreasing on a b then 1 n Error in Left n Right n Error in Mid n Trap n Error in Simp n 1 n2 1 n4 If f is ccd on a b then If f is ccu on a b then Simpson s Rule 7 6 Improper Integrals Occur when trying to integrate functions whose graphs are infinite in extent either horizontally vertically or both Type 1 Horizontally Infinite Region Let f x be defined for If exists and is finite converges So diverges Otherwise we say Type 2 Vertically Infinite Region Suppose f x is defined on a b and that or as vertical asymptotes If exists and is finite then converges and diverges Otherwise Type 3 Vertically and Horizontally infinite region basically a combination of types I and II which we handle by breaking up the interval of integration 7 7 Comparison of Improper Integrals The Comparison Test for Let 1 Guess by looking at the integrand for large x what the integrand behaves like and whether it converges or not This gives us a fn to compare the integrand with 2 Compare the guess by comparison If and converges then converges If and diverges then diverges Useful Integrals to Compare With Limit Comparison Test Type I If f x g x are continuous and positive on and Then exists with and either both converge or both diverge Chapter 8 Using the Definite Integral 8 1 Areas and Volumes Areas by Horizontal Slices width w w h of horizontal cross section depends on height h Add all pieces together to get approx total area A Areas by Vertical Slices Volumes by Horizontal Slices Volumes by Vertical Slices 8 2 Applications to Geometry To Compute a Volume Area Length Using an Integral Divide the solid region length into small pieces whose volume area length we can easily approximate Add the contributions of the pieces together gives us a Riemann Sum which approximates the total volume area length Take the limit as the number of terms in the sum tends to infinity Leads to a definite integral for the total volume area length Volumes of Revolution Arc Length 8 3 Polar Coordinates Conversion of Polar Coordinates to Cartesian Coordinates Conversion of Cartesian Coordinates to Polar Coordinates x2 y2 r2 r2 x2 y2 So and not sufficient to tell in which quadrant should be Use angle measure to find correct quadrant What if x 0 doesn t make sense But is y 0 then we need a value of which puts us on the positive y axis Finally if x y 0 and we re at the origin then r 0 and is undefined If y 0 then we need a value of which puts us on the negative y axis Except for this case given x y all possible values of differ by integer multiples of 2 which extends from the origin like in the example above Curves in Polar Coordinates i r 2 This is just a circle of radius 2 about the origin ii This is a half line with slope iii y 1 This is a horizontal line in Cartesian Coords iv r 2cos This is a circle of radius 1 centered at 1 0 v r 1 cos This curve is called a cardoid vi r 3sin 2 This curve is called a four leaved rose vii r2 a2cos 2 This curve is called a lemniscate Area in Polar Coordinates Slope in Polar Coordinates Arc Length in Polar Coordinates 8 4 Density and Center of Mass To find the total amount of some quantity from density 1 Divide the region into small pieces of approx constant density The amount for each piece is approx density x size of piece 2 Add up all the contributions for all pieces Gives us Riemann sum approximating the total amount 3 Let the number of pieces tend to in a suitable way Gives us an integral which we define to be total amount Center of Mass One Dimensional Region where M is the total mass Two Dimensional Region Three Dimensional Region 8 5 Applications to Physics Work Energy W F d Units Force lb Work ft lb English Units Work Against a Variable Force Force Newton 1N 1kg m s2 Work Energy Joule 1J 1Nm kg m2 s2 Hooke s Law k spring constant L natural length L extended length Force Pressure F P A Metric Unit of Pressure Pascal Pa 1Pa 1 N m2 English Unit lbs ft2 or lbs in2 psi Hydrostatic Pressure acts equally in all directions and at a depth n below the surface pressure is given by p density of fluid g acceleration due to gravity 8 7 8 8 Distribution Functions Densities Probability Mean Median Probability Density Function pdf A function p x is a pdf for some quantity x associated with a population if Cumulative Distribution Function cdf The fraction of the population for which x is less than or equal to some value t Using the density p x this gives us a fn P t defined by P t is the antiderivative of p t The Median OR The Mean Chapter 9 Sequences and Series 9 1 Infinite Sequences Series a sum of numbers Sequences list of ordered numbers terms A common way of writing a sequence is Fibonacci Sequence 1 1 2 3 5 8 13 21 34 55 S1 1 S2 1 S3 Sn 1 Sn 2 for Recursive the formula for each term involves the previous terms Convergence of Series Important Examples I Sn xn If x 1 then If x 1 then If x 1 then If …