6 Integration Antiderivatives and the Rules of Integration Integration by Substitution Area and the Definite Integral The Fundamental Theorem of Calculus Evaluating Definite Integrals Area Between Two Curves Applications of the Definite Integral to Business and Economics Volumes of Solids of Revolution

Integral (积分): Antiderivative (反导数) In section 2.4, given position (位置), find its velocity (速度)—Derivative Opposite problem now: given velocity, find position—Integral

Integral (积分): Antiderivative (反导数) Definition: A function F is an antiderivative of f on an interval I if F (x) = f(x) for all x  I. Example: F(x) = x3 − 2 x2 + x − 1 is an antiderivative of f(x) = 3 x2 − 4 x + 1 Remember: (xn)  = n xn−1 , (c)  = 0 Example: Let F(x) = x, G(x) = x + 2, and H(x) = x + c, (constant c). All antiderivatives of f(x) = 1.

Indefinite (不定的) Integral (积分) Antidifferentiation (反微分) or Integration: : integral sign（积分符号） f(x) : integrand（被积函数） c : constant of integration（积分常数） x: independent variable（自变量） If independent variable is t, then Both t and x are dummy (虚拟) variables

Basic Integration Rules Rule 1: , where k a constant Check F  (x) = (kx + c)  = k + 0 Example:  dx =  x + C Rule 2: Power Rule

Basic Integration Rules Rule 3: Integral of a constant multiple of a function

Basic Integration Rules Rule 4: The sum rule

Basic Integration Rules Rule 5: Exponential function

Basic Integration Rules

Integration by Substitution (换元法) Example: Method 1：expand (2x + 4)50，then use rules Method 2 : change of variable

Integration by Substitution Detailed process: g(x) = 2 x + 4, f( t ) = t50, f( g(x) ) = (2x + 4)50

Integration by Substitution (换元法) Technique for Substitution T1: substitute inside function T2: substitute higher degree factor