微积分英文课件：Chapter 4 Integration (2).ppt

Chapter 4 Integration 4.1 Indefinite Integrals  Indefinite Integrals An indefinite integral of f(x) represent an entirely family of functions whose derivative is f(x) and is denoted by Example 1: Table of Indefinite integrals The properties of Indefinite Integrals Example 3: Example 4: Example: 4.2 The Substitution Rule The Substitution Rule The Substitution Rule: The Substitution Rule: 4.3 Estimating with finite sums 2. Distance traveled Suppose v=v(t) is continuous on [a,b]. How far the object traveled on the time interval [a,b] ? 4.4 The definition of integral The definite integral as a limit of Riemann sums Let f be a function defined on a closed interval [a,b]. For any partition P of [a,b], let the number ck be chosen arbitrarily in the subinterval [xk-1,xk]. If there exists a number I such that no matter how P and the ck are chosen ,the f is integrable on [a,b] and I is the the definite integral of f over [a,b]. Caution: The definite integral is a number; it does not depend on x. In fact ,we could use and letter in place of x without changing the value of the integral: The Midpoint Rule Properties of the definite integral When we defined the definite integral . we implicitly assumed that ab .But the definition as a limit of Riemann sums makes sense even if ab. Notice that if we reverse a and b ,then change from (b-a)/n to (a-b)/n . Therefore Properties of the integral Example: Use the properties of integrals to evaluate 4.5 The Fundamental Theorem of Calculus The Fundamental Theorem of Calculus The fundamental Theorem of Calculus is appropriately named because it establishes a connection between the two branches to Calculus: differential calculus and integral calculus. Differential calculus arose from the tangent problem, whereas integral calculus arose from a seemingly unrelated problem ,the area problem. The Fundamental Theorem of Calculus gives the precise relationship between the deriv