Download presentation
Presentation is loading. Please wait.
1
Applications of the Definite Integrals Dr. Faud Almuhannadi Math 119 - Section(4)
2
Done by: Hanen Marwa Najla Noof Wala
3
In this part, we are going to explain the different types of applications related to the “ Definite Integrals “. Which includes talking about : 1. Area under a curve 2. Area between two curves 3. Volume of Revolution
4
Definition : In calculus, the integral of a function extends the concept of an ordinary sum. While an ordinary sum is taken over a discrete set of values, integration extends this concept to sums over continuous domainscalculusfunction sumdiscretecontinuousdomains
5
The simplest case, the integral of a real- valued function f of one real variable x on the interval [a, b], is denoted:
![The simplest case, the integral of a real- valued function f of one real variable x on the interval [a, b], is denoted:](http://images.slideplayer.com/17/5361062/slides/slide_5.jpg "The simplest case, the integral of a real- valued function f of one real variable x on the interval [a, b], is denoted:")
6
The ∫ sign represents integration ; a and b are the lower limit and upper limit of integration, defining the domain of integration; f(x) is the integrand; and dx is a notation for the variable of integration
7
Computing integrals The most basic technique for computing integrals of one real variable is based on the fundamental theorem of calculus. It proceeds like this:
8
Choose a function f(x) and an interval [a, b]. Find an antiderivative of f, that is, a function F such that F' = f. By the fundamental theorem of calculus, provided the integrand and integral have no singularities on the path of integration, Therefore the value of the integral is F(b) − F(a).
![Choose a function f(x) and an interval [a, b].](http://images.slideplayer.com/17/5361062/slides/slide_8.jpg "Find an antiderivative of f, that is, a function F such that F = f. By the fundamental theorem of calculus, provided the integrand and integral have no singularities on the path of integration, Therefore the value of the integral is F(b) − F(a)..")
9
Case..1.. Area Under a Curve
10
Example..1.. The graph below shows the curve and is shaded in the region
11
The area is found by integrating
12
Example..2..
13
Case..2.. Area between two curves
14
Say you have 2 curves y = f(x) and y = g(x)
15
Area under f(x)= Area under g(x)=
16
Superimposing the two graphs: Area bounded by f(x) and g(x)
17
Example..3.. Find the area between the curves y = 0 and y = 3(x 3 - x)
19
Example..4.. Find the area bounded by the curves y = x 2 - 4x – 5 and y = x + 1
20
Solving the equations simultaneously, x + 1 = x 2 - 4x - 5 x = -1 or x = 6 Required Area =
21
Volume Of A Revolution
22
A solid of revolution is formed when a region bounded by part of a curve is rotated about a straight line. Rotation about x-axis:
23
Rotation about y-axis:
24
Example..5.. The volume that we are looking for is shown in the diagram below
25
To find the volume, we integrate
26
Thank u 4 listening
Similar presentations
![6.5 The Definite Integral In our definition of net signed area, we assumed that for each positive number n, the Interval [a, b] was subdivided into n subintervals.](/15/4748588/big_thumb.jpg "6.5 The Definite Integral In our definition of net signed area, we assumed that for each positive number n, the Interval [a, b] was subdivided into n subintervals.>")
![8 TECHNIQUES OF INTEGRATION. In defining a definite integral, we dealt with a function f defined on a finite interval [a, b] and we assumed that f does.](/16/5012710/big_thumb.jpg "8 TECHNIQUES OF INTEGRATION. In defining a definite integral, we dealt with a function f defined on a finite interval [a, b] and we assumed that f does.>")
![1 Fundamental Theorem of Calculus Section 4.4. 2 The Fundamental Theorem of Calculus If a function f is continuous on the closed interval [a, b] and F.](/19/5780760/big_thumb.jpg "1 Fundamental Theorem of Calculus Section 4.4. 2 The Fundamental Theorem of Calculus If a function f is continuous on the closed interval [a, b] and F.>")
![Definition: the definite integral of f from a to b is provided that this limit exists. If it does exist, we say that is f integrable on [a,b] Sec 5.2:](/21/6236247/big_thumb.jpg "Definition: the definite integral of f from a to b is provided that this limit exists. If it does exist, we say that is f integrable on [a,b] Sec 5.2:>")
