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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

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 :

  • Area under a curve

  • Area between two curves

  • Volume of Revolution

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

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

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

The most basic technique for computing integrals of one real variable is based on the fundamental theorem of calculus. It proceeds like this:

  • 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..

Area Under a Curve

Example ..1..

The graph below shows the curve

and is shaded in the region

The area is found by integrating

Example ..2..

Case ..2..

Area between two curves

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

  • Area under f(x)=

  • Area under g(x)=

Superimposing the two graphs:

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

Example ..3..

  •  Find the area between the curves

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

Example ..4..

  • Find the area bounded by the curves

    y = x2 - 4x – 5


    y = x + 1

  • Solving the equations simultaneously,

              x + 1 = x2 - 4x - 5

               x = -1 or x = 6

    Required Area =

Volume Of A Revolution

  • 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:

Rotation about y-axis:

Example ..5..

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

  • To find the volume, we integrate

Thank u 4 listening

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