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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

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

### 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

and

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: