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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)

- 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

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

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).

Area Under a Curve

The graph below shows the curve

and is shaded in the region

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)

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

- 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:

Rotation about y-axis:

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

- To find the volume, we integrate

Thank u 4 listening