Applications of the definite integrals
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Applications of the Definite Integrals. Dr. Faud Almuhannadi Math 119 - Section(4). Done by:. Hanen Marwa Najla Noof Wala. In this part, we are going to explain the different types of applications related to the “ Definite Integrals “. Which includes talking about :

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Applications of the Definite Integrals

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Applications of the definite integrals

Applications of the Definite Integrals

Dr. Faud Almuhannadi

Math 119 - Section(4)


Done by

Done by:

  • Hanen

  • Marwa

  • Najla

  • Noof

  • Wala


Applications of the definite integrals

In this part, we are going to explain the

different types of applications related to the “ Definite Integrals “.

Which includes talking about :

  • Area under a curve

  • Area between two curves

  • Volume of Revolution


Definition

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 continuousdomains


Applications of the definite integrals

The simplest case, the integral of a real-valued function f of one real variable x on the interval [a, b], is denoted:


Applications of the definite integrals

The ∫ sign represents integration; aandb 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


Computing integrals

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:


Applications of the definite integrals

  • 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).


Case 1

Case ..1..

Area Under a Curve


Example 1

Example ..1..

The graph below shows the curve

and is shaded in the region


The area is found by integrating

The area is found by integrating


Example 2

Example ..2..


Case 2

Case ..2..

Area between two curves


Applications of the definite integrals

Say you have 2 curves y = f(x) and y = g(x)


Applications of the definite integrals

  • Area under f(x)=

  • Area under g(x)=


Applications of the definite integrals

Superimposing the two graphs:

Area bounded by f(x) and g(x)


Example 3

Example ..3..

  •  Find the area between the curves

           y = 0      and      y = 3(x3 - x)


Example 4

Example ..4..

  • Find the area bounded by the curves

    y = x2 - 4x – 5

    and

    y = x + 1


Applications of the definite integrals

  • Solving the equations simultaneously,

              x + 1 = x2 - 4x - 5

               x = -1 or x = 6

    Required Area =


Applications of the definite integrals

Volume Of A Revolution


Applications of the definite integrals

  • 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:


Applications of the definite integrals

Rotation about y-axis:


Example 5

Example ..5..

  • The volume that we are looking for is shown in the diagram below


Applications of the definite integrals

  • To find the volume, we integrate


Applications of the definite integrals

Thank u 4 listening


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