Math Dr. Melahat Almus. OFFICE HOURS (610 PGH) MWF 9-9:45 am, 11-11:45am, OR by appointment.

Transcription

1 Math 43 Dr. Melahat Almus OFFICE HOURS (60 PGH) MWF 9-9:45 am, -:45am, OR by appointment. COURSE WEBSITE: Visit my website regularly for announcements and course material! If you me, please mention your course (43) in the subject line. BUBBLE IN PS ID VERY CAREFULLY! If you make a bubbling mistake, your scantron will not be saved in the system and you will not get credit for it even if you turned it in. Bubble in Popper Number. Be considerate of others in class. Respect your friends and do not distract anyone during the lecture. Did you reserve a seat for Test 4?

2 POPPER# Question# 6x x? A) 5 B) C) 0 D) 6 E) None /4 Question# 0 sec x? A) B) sqrt() C) / D) E) None

3 Section 6.3 Basic Integration Rules TABLE OF INTEGRALS r r x x C ; r r ln x C x sin x cos x C cos x sin x C sec x tan x C csc x cot x C sec xtan x sec x C csc x cot x csc x C x x e e C x x a a C ; a 0, a. ln a sinh x cosh x C cosh x sinh x C x arcsin x C x arctan x C 3

4 Integrals Involving Absolute Value x 4 x 5 x 4 4

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6 Exercise: 5 x 3x4 5

7 Applications If the rate at which a quantity is changing is given, we can find the net change on the function by using integrals. For example, if the rate at which the volume is changing is given, then the net change on the volume is found by integrating that rate. If the rate at which an object is moving (speed) is given, we can find the total distance covered by that object by integrating the speed. Recall: a: acceleration, v: velocity, s: position function. ' and vt s t at v' t s''( t) If the acceleration is given, we can find the velocity and position functions using integration. b a vt at dt b a st vt dt 6

8 Total displacement vs Total Distance Covered: t v t dt ststgives the total displacement of the object from time t t t t vt dt total distance traveled from time t t t to t t. to t t. 7

9 Example: An object moves along a line with acceleration at 6t 6, where t 0 represents the time. Initial velocity is 0 and initial position of the object is units left of the origin. Find the velocity of this object after 4 seconds. Find the position function. Find the total displacement of this object in the first 4 seconds. 8

10 Find the total distance covered by this object in the first 4 seconds. Exercise: Water is poured into a tank at a rate of rt 6t cubic feet per second. How much water is poured into the tank in the first seconds? 9

11 Example: Given f '( x) 4x, f() 5, find f ( x ). Example: Given f ''( x) 6x, f '(0), f(0) 0, find f ( x ). 0

12 Section 6.4 Integration by Substitution Example: x Example: x 5 The method of substitution is based on the Chain Rule: f g' x f' g x g' x. When we have an integrand of the form this process: g' x f ' g x, we can find the integral by reversing f ' g x g' x f g x. In the integral, we let u gx and then the differential of u is du g' x. Hence, f ' g x g' x f' u du f u C f g x C. gx by u and substitution g' x by du is called substitution (or u substitution). We use Replacing this method for when the integrand is a composition of functions.

13 Example: x x 5 3 Example: x x x

14 Example: 0x x 3 Example: x x 3

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16 Example: e 4x Example: xe x 4