Chapter 5 Integrals. 5.1 Areas and Distances

Transcription

1 Chapter 5 Integrals 5.1 Areas and Distances We start with a problem how can we calculate the area under a given function ie, the area between the function and the x-axis? If the curve happens to be something easy like a horizontal line - this isn t too hard, but if it s a curve, the problem becomes more difficult. Let s start by drawing a rectangle that s f(a) tall by (b a) wide as an estimate: Area = f(a) (b a) How can we get a better estimate? Let s try 2 rectangles: Area = f(a) x 1 + f(c) x 2 How could we get an even better estimate?

2 Ex: Estimate the area under the curve f(x) = x 2 between x = 0 and x = 1 by constructing 8 rectangles (the book calls them strips ) of equal width. This seems like a relatively simple situation, until you realize that you get different answers depending on whether you draw the rectangles from the left or from the right: Using left endpoints, we get: f(0) " + f " " + f # " " + f $ " " + f % " " + f & " " + f ' " " + f ( " " We can see this will be a lower bound, because all the rectangles fit completely under the curve. Using right endpoints, we get: f " " + f # " " + f $ " " + f % " " + f & " " + f ' " " + f ( " " + f 1 " We can see this will be an upper bound, because all the rectangles extend to above the curve. By computing both, we get: A Hmmm if only we had a technique for looking at a progression of smaller and smaller rectangles.

3 Definition: The Area A of the region S that lies under the graph of the continuous function f is the limit of the sum of the areas of approximating rectangles (using right endpoints): A = lim - / R - = lim - / f x x + f x # x + + f(x - ) x 1) It can be proven that this limit always exists. 2) It can also be shown that you get the same limit value A from using left endpoints. 3) In fact, you can choose any x-value in each interval to calculate the height. The values that are chosen - x 1 *, x 2 *,, x n * - are called the sample points. Using sigma notation we get: A = lim - / - :; f(x : ) x (Riemann sum) The Distance Problem My odometer is broken, but I d still like to calculate how far I ve driven using my speedometer and a stopwatch (held by my passenger, for safety.) I know D = R*T and I have the following data: Time (s) Speed (ft/s) How can I estimate the distance I ve travelled?

4 Using Left Endpoints: 0 * * * 5 = 225 ft Using Right Endpoints: 20 * * * 5 = 275 ft 1) How can we improve our estimate? 2) Do we know whether the true distance lies between the two estimates?

5 5.2 The Definite Integral The limit of the sum from 5.1: lim f x x + f x # x + + f(x - ) x = lim - / - / - :; f(x : ) x where x = = -, can be written as: f x dx and is called the definite integral of f from a to b, as long as the limit exists and is the same for all choices of x i. If the limit exists, then f is said to be integrable over [a,b] Properties of the Integral 1) c dx = c(b a) where c is any constant 2) f x + g x dx = f x dx + g x dx 3) c f x dx = c f x dx where c is any constant 4) [ f x g x ]dx E 5) f x dx + f x dx E = f x dx = f x dx 6) If f(x) 0 for a x b, then f x dx 7) If f(x) g(x) for a x b, then f x dx g x dx 8) If m f(x) M for a x b, then m (b a) f x dx M(b a) - g x dx 0

6 Ex: (#33) The graph of f is shown. Evaluate each integral by interpreting it in terms of areas. # a) f x dx d) f x dx K

7 5.3 The Fundamental Theorem of Calculus (Cue exciting music) The Fundamental Theorem of Calculus, Part I If f is continuous on [a,b], then the function g defined by g(x) = M f t dt a x b is continuous on [a,b] and differentiable on (a,b), and g (x) = f(x). Ex: Find N NM M R sec t dt (Hint: You ll need the Chain Rule) The Fundamental Theorem of Calculus, Part 2 If f is continuous on [a, b], then f x dx = F(b) F(a) where F is any antiderivative of f. That is, a function s. t. F = f. Why can it be *any* antiderivative of f?

8 So, differentiation and integration are inverse processes. Ex: Evaluate these integrals 1) 1 + x + 3x # dx % 2) 4 t t dt Note: Be careful when rushing in to evaluate integrals. Look at this example: $ = 1 x # dx It seems like it would be easy to find an antiderivative and evaluate it, but first is this function continuous over [-1, 3]?

9 5.4 Indefinite Integrals Definite vs. Indefinite Integrals: $ 2x dx number: is a definite integral. It s a string of symbols that represent a $ 2x dx = x 2 ] 0 3 = = 9 2x dx is an indefinite integral. It s a string of symbols that represent a function, or family of functions: 2x dx = x 2 + C Table of Indefinite Integrals a f x dx = a f(x) dx [f(x) + g(x)] dx = f x dx + g x dx kdx = kx + C x n dx = MXYZ -[ + C (n -1) e x dx = e x + C M dx = ln x + C cos x dx = sinx + C sin x dx = - cos x + C sec 2 x dx = tan x + C sec x tan x dx = sec x + C

10 Net Change Theorem The integral of a rate of change is the net change F ] x dx = F b F(a)

11 5.5 The Substitution Rule Ex: a) Find N NM (x# + 3x) $ b) Find x # + 3x # 6x + 9 dx The Substitution Rule If u = g(x) is a differentiable function whose range is an interval I and f is continuous on I, then f g x g ] x dx = f u du Ex: Evaluate 2x + 1 dx using the Substitution Rule

12 Ex: Evaluate 1 + x # x & dx using the Substitution Rule Integrals of Symmetric Functions If f is even [ f(-x) = f(x) ], then If f is odd [ f(-x) = -f(x) ], then = = f x dx f x dx = 0 = 2 f x dx