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Applications of the Definite Integrals PowerPoint Presentation

Applications of the Definite Integrals

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Presentation Transcript

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 function ∫ 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 function

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 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.. function

Area Under a Curve

The area is found by integrating function

Example ..2.. function

Case ..2.. function

Area between two curves

- Area under f(x)= function
- Area under g(x)=

Superimposing the two graphs: function

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

Example ..3.. function

- Find the area between the curves
y = 0 and y = 3(x3 - x)

Example ..4.. function

- Find the area bounded by the curves
y = x2 - 4x – 5

and

y = x + 1

- Solving the equations simultaneously, function
x + 1 = x2 - 4x - 5

x = -1 or x = 6

Required Area =

Volume Of A Revolution function

- 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: part of a curve is rotated about a straight line.

Example ..5.. part of a curve is rotated about a straight line.

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

- To find the volume, we integrate part of a curve is rotated about a straight line.

Thank u 4 listening part of a curve is rotated about a straight line.

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