By its nature, Calculus can be intimidating. This method is especially useful when you need to integrate by parts more than once to solve a problem. P, The Product Rule: d/dx (f(x)g(x)) = f '(x)g(x) + f(x)g '(x), The Quotient Rule: d/dx (f(x)/g(x)) = (f '(x)g(x) - f(x)g '(x))/(g(x), d/dx (f(u(x))) = d/dx (f(u)) d/dx (u(x)) + f (3) (a)(x-a) 3 /3! By its nature, Calculus can be intimidating. a: ( ) 69 0 obj <>/Filter/FlateDecode/ID[]/Index[45 47]/Info 44 0 R/Length 114/Prev 162259/Root 46 0 R/Size 92/Type/XRef/W[1 3 1]>>stream When doing Calculus, the formula for integration by parts gives you the option to break down the product of two functions to its factors and integrate it in an altered form. I may keep working on this document as the course goes on, so these notes will not be completely finished until the end of the quarter. f(a) + f '(a)(x -a) + f ''(a)(x-a) 2 /2! Various Formulas 118 B.1. 0 12 2. cos ( ) 1 cos(2 )x x. The table below shows you how to differentiate and integrate 18 of the most common functions. Basic Properties and Formulas If fx and g x are differentiable functions (the derivative exists), c and n are any real numbers, 1. cf cf x 2. fg fx gx 3. fg f g fg – Product Rule 4. Here it is formally: The Constant Multiple Rule for Integration tells you that it’s okay to move a constant outside of an integral before you integrate. Arc Length Surface Area : Note that this is often a Calc II topic. Some integrals of compound functions f (g(x)) are easy to do quickly in Calculus. MATHEMATICS – USEFUL FORMULAE COORDINATE GEOMETRY Straight Line Equation y −y 1 = m(x −x 1) Circle (x −a)2 +(y −b)2 = r2 has a centre (a,b) and radius r Parabola y2 = 4ax or (at2,2at) Focus (a,0) Directrix x = −a Ellipse x2 a2 + y2 b2 =1 or (acosθ,bsinθ) Foci (c,0) (–c,0) where b2 = a2 – c2 Eccentricity: e = c a Hyperbola x2 a2 − y2 b2 =1 or (asecθ,btanθ) asymptotes y = ± b + f (4) (a)(x-a) 4 /4! and then isolate all x variables on one side of the equal sign. If D(a,b) < 0 then f has a saddle point at P. If D(a,b) = 0 then the test is inconclusive. �� "Mg��� `s�@�L��� V h�b```a`` These include compound functions for which you know how to integrate the outer function f, and the inner function g(x) is of the form ax — that is, it differentiates to a constant. If there are no sec(x) factors and the power of tan(x) is even and positive, use sec 1 tan22x x. to convert one tan2xto sec2x. To use integration by parts in Calculus, follow these steps: Decompose the entire integral (including dx) into two factors. Let f be continuous with f (x) ≥ 0 on the interval [a, b]. The three basic formulas are, b a L ds=∫ 2 b a SA yds=∫ π (rotate about x-axis) 2 b a SA xds=∫ π (rotate about y-axis) () ( ) ( ) ()( ) ( ) ()( ) ( ) ( ) →∞ ∞ ∫∫ ( ) −∞ →−∞ ∫∫ = ( ) ( ) ( ) −− ∞ ∞ ∞ ∞ ∫ ∫∫ =+ provided BOTH integrals are convergent. Make another substitution to change dx and all other occurrences of x in the integral to an expression that includes du. 0 d c dx Assume that f(x,y) is a nonnegative on R. Then the volume under the These include compound functions for which you know how to integrate the outer function f, and the inner function g(x) is of the form ax + b — that is, it differentiates to a constant. graph of z = f(x,y) above R is given by, Suppose f(x,y) is a function and R is a region on the xy-plane. Suppose f(x,y) is a function and R is a region on the xy-plane. = f'(u)u'(x), Simpson's Rule (Parabolic Approximation) for ∫. When figuring Calculus problems, some integrals of compound functions f (g(x)) are easy to do quickly. But you can take some of the fear of studying Calculus away by understanding its basic principles, such as derivatives and antiderivatives, integration, and solving compound functions. Summation Formulas 118 Appendix C. Table of Integrals 119. Integrate by using u as your new variable of integration. h��X[O�:�+~��/� The Sum Rule for Integration tells you that it’s okay to integrate long expressions term by term. h�bbd```b``f�� �q�d�fk��`RD�փH��`2,�fK��V0i Then the AVERAGE VALUE of z = f(x,y) over the region R is given by. + ... + f (n) (a)(x-a) n /n! to allow the number of rectangles to approach infinity: The DI-agonal method is basically integration by parts with a chart that helps you organize information. Infinite Series The sum of the Geometric Series You get r from measuring the distance of f (x) to the x-axis. Introduction These notes are intended to be a summary of the main ideas in course MATH 214-2: Integral Calculus. But you can take some of the fear of studying Calculus away by understanding its basic principles, such as derivatives and antiderivatives, integration, and solving compound functions. Calculus II For Dummies Cheat Sheet. Variable substitution allows you to integrate when the Sum Rule, Constant Multiple Rule, and Power Rule don’t work. The Riemann Sum formula is as follows: Below are the steps for approximating an integral using six rectangles: Increase the number of rectangles (n) to create a better approximation: Simplify this formula by factoring out w from each term: Use the summation symbol to make this formula even more compact: The value w is the width of each rectangle: Each h value is the height of a different rectangle: So here is the Riemann Sum formula for approximating an integral using n rectangles: For a better approximation, use the limit. dx is the thickness or width of the disk. sin ( ) 1 cos(2 )x x. ii. ADVANCED PLACEMENT PHYSICS 2 EQUATIONS, EFFECTIVE 2015 CONSTANTS AND CONVERSION FACTORS Proton mass, 1.67 10 kg 27 m p =¥-Neutron mass, 1.67 10 kg 27 m n =¥-Electron mass, 9.11 10 kg 31 m e =¥-Avogadro’s number, 23 -1 N 0 =¥6.02 10 mol Universal gas constant, R =8.31 J (mol K) i Boltzmann’s constant, 1.38 10 J K. 23. k. B =¥-Electron charge magnitude, e =¥1.60 10 C Also discover a few basic rules applied to Calculus like Cramer’s Rule, and the Constant Multiple Rule, and a few others, and you’ll be on your way to acing the course. endstream endobj startxref Rules for sec(x) and tan(x) also … Let the factor without dx equal u and the factor with dx equal dv. Given a function f(x) the Taylor Polynomial 2 ffgfg gg – Quotient Rule 5. 91 0 obj <>stream Use the following table for integration by parts using the DI-agonal method: When you perform integration, there are three important rules that you need to know: the Sum Rule, the Constant Multiple Rule, and the Power Rule. Differentiate u to find du, and integrate dv to find v. Evaluate the right side of this equation to solve the integral. Here it is expressed in symbols: The Power Rule for Integration allows you to integrate any real power of x (except –1). If the region R bounded by the graph of f, the x-axis, and the lines x = a and x = b is revolved about the x-axis, the volume of the resulting solid of revolution is: Area of circle = πr 2. endstream endobj 46 0 obj <> endobj 47 0 obj <> endobj 48 0 obj <>stream As you can see, integration reverses differentiation, returning the function to its original state, up to a constant C. The Riemann Sum formula provides a precise definition of the definite integral as the limit of an infinite series. Here’s the Power Rule expressed formally: In Calculus, you can use variable substitution to evaluate a complex integral. 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