Science One Math. January

Transcription

1 Science One Math January

2 (last time) The Fundamental Theorem of Calculus (FTC) Let f be continuous on an interval I containing a. 1. Define F(x) = f t dt with F (x) = f(x). on I. Then F is differentiable on I 2. Let G be any antiderivative of f on I. Then for any b in I ' f t dt = G b G(a)

3 FTC part I: The area function if f(x) is continuous on I let F(x) = area under f(x) on [a, x] F(x) = f t dt F(x) is called accumulation function

4 In science there are many functions defined as an integral: Error function Erf(x) = 0 1 e 34 5 dt 6 (probability and statistics) Sine integral function Si(x) = dt (signal processing) Fresnel functions S(x) = sin t 0 dt 6 C(x) = cos t 0 6 dt (theory of diffraction) Natural logarithm ln(x) =?? dt 4 for x > 0

5 Problem: Let F(x) = t D dt. Considering the interval [-1, 1], find 3? where F is increasing? decreasing? Where does F have local extrema?

6 Example: Let F(x) = t D dt. Considering the interval [-1, 1], find where F is 3? increasing? decreasing? Where does F have local extrema? Recall: FTC part I : If F(x) = f t dt F x = f x., F is differentiable on [a, x] with

7 Problem: Let F(x) = t D dt. Considering the interval [-1, 1], find where F is 3? increasing? decreasing? Where does F have local extrema? Recall: FTC part I : If F(x) = f t dt f x., F is differentiable on [a, x] with F x = Solution: F is increasing when F > 0. By FTC, F x = x D x 3 > 0 when x > 0 F is increasing on (0,1), decreasing on (-1,0). F(0) is a local minimum, F(0) = 6 3? t D dt =? We need to compute the definitive integral. Let s use the second part of FTC.

8 FTC part II: f t dt = G b G(a), where G (x)=f(x). Proof: If F(x) = f t dt, we know F (x) = f(x). That is, F is an antiderivative of f. Suppose G is any antiderivative of f on I. Then G(x) = F(x) + C for some constant C. If x = a, F(a) = 0 so G a = C. If x = b, F b = f t dt G b = F b + G a = f t dt ' f t dt = G b G a. + G(a).

9 so now we know how to compute definite integrals without Riemann sums! Previous problem: Let F(x) = 3? t D dt. Compute F(0). F(0) = 6 3? t D dt =? L tl 0 1 = 0 (3?)N L =? L

10 Recap: the derivative undoes the integral, and vice versa! d dx ' f t dt = f(x) ' F O x dx = F b F a Evaluating integrals involves finding antiderivatives instead of integration, we should call it antidifferentiation!

11 Finding antiderivatives may be hard! A few techniques to find antiderivatives Substitution Integration by parts Partial fractions Trigonometric integrals Trigonometric substitutions

12 definite vs indefinite integral The definite integral: f x dx The indefinite integral: f x dx This is a number! This is a family of functions!

13 Problems: 1) Find the area of the region under y = 3x x 2 and above x-axis. [Ans: 276] 2) Find the area of the region under y= P 5 Q? and above y = 1. [Ans: 10arctan(2)-4] 3) Find 4) Find R P R x0 e 340 dt 3L R D R e 340 dt 5) If f t dt =0 and f is continuous on [a, b], prove there is a point c in [a,b] with f(c) = 0.

14 Other interpretations of the definite integral F O x dx = F b F a The integral of a rate of change is the net change.

15 Other interpretations of the definite integral F O x dx = F b F a The integral of a rate of change is the net change. (Physics) Let s(t) be position function of an object that moves along a line with velocity v(t), then 4 0 v t dt = s t 4 2 s t 1 is the net change in position.?

16 Other interpretations of the definite integral F O x dx = F b F a The integral of a rate of change is the net change. (Physics) Let s(t) be position function of an object that moves along a line 4 with velocity v(t), then 0 v t dt = s t 4 2 s t 1 is the net change in? position. (chemistry) Let [C](t) be the concentration of the product of a chemical reaction at time t, then 4 0 R[X] dt = C t 4 2 [C] t 1 is net change in concentration.? R4

17 Other interpretations of the definite integral F O x dx = F b F a The integral of a rate of change is the net change. (Physics) Let s(t) be position function of an object that moves along a line with 4 velocity v(t), then 0 v t dt = s t 4 2 s t 1 is the net change in position.? (chemistry) Let [C](t) be the concentration of the product of a chemical reaction 4 at time t, then R[X] 0 dt = C t 4? R4 2 [C] t 1 is net change in concentration. (biology) Let p(t) be a population size at time t, then 4 0 RZ dt R4 = p t 4 2 p t 1 is net change in population size.?

18 Applications of the integral Areas between curves Volumes of solids Work done by non constant force Average value of a function Arc length Surface area of solids Hydrostatic pressure Probability density functions Centre of mass..and more

19 Areas between curves The area of a region bounded by the curves y = f(x) and y = g(x) and the lines x = a and x = b is '[f x g x ]dx where f and g are continuous and f(x) g(x) for all x in [a, b].

20 Problems 1) Find the area of the region enclosed by the parabola y = 2 x 2 and the line y = x. 2) Find the area of the region in the first quadrant that is bounded above by y = x and below by the x-axis and the line y = x 2. 3) Redo problem 2 by integrating with respect to y. 4) Find the total area A lying between the curves y = sin x and y = cos x from x = 0 to x = 2π.

21 Compute the area enclosed by the heart-shaped curve (from February 14, 2017 midterm) y = x 4 if 4 x 4 y = 0.3x 4 x + 2 if 0 x 4 y = 0.3x 4 x + 2 if 4 x 0 x = 0.3y y + 4 if 0 y 2 x = 0.3y 0 0.6y 4 if 0 y 2