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Applications of the Definite Integrals Dr. Faud Almuhannadi Math 119 - Section(4)
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Done by: Hanen Marwa Najla Noof Wala
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In this part, we are going to explain the different types of applications related to the “ Definite Integrals “. Which includes talking about : 1. Area under a curve 2. Area between two curves 3. Volume of Revolution
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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 continuous domainscalculusfunction sumdiscretecontinuousdomains
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The simplest case, the integral of a real- valued function f of one real variable x on the interval [a, b], is denoted:
![The simplest case, the integral of a real- valued function f of one real variable x on the interval [a, b], is denoted:](http://images.slideplayer.com./17/5361062/slides/slide_5.jpg "The simplest case, the integral of a real- valued function f of one real variable x on the interval [a, b], is denoted:")
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The ∫ sign represents integration ; a and b 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
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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:
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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).
![Choose a function f(x) and an interval [a, b].](http://images.slideplayer.com./17/5361062/slides/slide_8.jpg "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)..")
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Case..1.. Area Under a Curve
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Example..1.. The graph below shows the curve and is shaded in the region
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The area is found by integrating
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Example..2..
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Case..2.. Area between two curves
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Say you have 2 curves y = f(x) and y = g(x)
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Area under f(x)= Area under g(x)=
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Superimposing the two graphs: Area bounded by f(x) and g(x)
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Example..3.. Find the area between the curves y = 0 and y = 3(x 3 - x)
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Example..4.. Find the area bounded by the curves y = x 2 - 4x – 5 and y = x + 1
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Solving the equations simultaneously, x + 1 = x 2 - 4x - 5 x = -1 or x = 6 Required Area =
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Volume Of A Revolution
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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:
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Rotation about y-axis:
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Example..5.. The volume that we are looking for is shown in the diagram below
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To find the volume, we integrate
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Thank u 4 listening
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