Engineering Mathematics: Application of Derivatives and Integrals

DIYALA UNIVERSITY COLLEGE OF ENGINEERING DEPARTMENT OF COMMUNICATIONS ENGINEERING Mathematics- I First Year Lecture 5 Asst. lecturer Wisam Hayder 2020 - 2021 1

Application of Derivatives Antiderivatives 2

Application of Derivatives 3

Application of Derivatives 4

Application of Derivatives 5

Application of Derivatives Indefinite Integrals 6

Application of Derivatives 7

INTEGRATION Area and Distance Area 8

INTEGRATION In each of our computed sums, the interval [a, b] over which the function is defined was subdivided into n subintervals of equal width (also called length) ? = (? ?) ? and was evaluated at a point in each subinterval: ?1 in the first subinterval, ?2in the second subinterval, and so on. The finite sums then all take the form. 9

INTEGRATION Distance Traveled Suppose we know the velocity function (t) of a car moving down a highway, without changing direction, and want to know how far it traveled between times t=a & t=b. If we already know an antiderivative F(t) of y(t) we can find the car s position function s(t) by setting s(t) = F(t) + C. The distance traveled can then be found by calculating the change in position, s(b) s(a) = F(b) F(a). 10

INTEGRATION Then we can approximate the distance traveled on each time subinterval with the usual distance formula distance = velocity time. and add the results across [a, b]. Suppose the subdivided interval looks like with the subintervals all of equal length ?. Pick a number ?1in the first interval, ?2is a number in the second interval. The distance traveled in the second time interval is about v(?2) ?. The sum of the distances traveled over all the time intervals is Where n is the total number of subintervals. 11

INTEGRATION The Definite Integral DDD 12

INTEGRATION 13

INTEGRATION 14

INTEGRATION 15

INTEGRATION Indefinite Integrals and the Substitution Method The set of all anti derivatives of a function is called indefinite integral of the function. Assume u and v denote differentiable functions of x, and a, n, and c are constants, then the integration formulas are:- 16

INTEGRATION 17

INTEGRATION 18

INTEGRATION 19

INTEGRATION 20

INTEGRATION Area Between Curves 21

INTEGRATION 22

INTEGRATION 23

Applications Of Definite Integrals Volumes Volumes Using Cylindrical Shells 24

Applications Of Definite Integrals

Applications Of Definite Integrals 26

Applications Of Definite Integrals 27

Applications Of Definite Integrals 28

Applications Of Definite Integrals Work 29

Applications Of Definite Integrals 30

Applications Of Definite Integrals 31