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Applications of the Definite Integrals

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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  1. Applications of the Definite Integrals Dr. Faud Almuhannadi Math 119 - Section(4)

  2. Done by: • Hanen • Marwa • Najla • Noof • Wala

  3. 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

  4. 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

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

  6. 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

  7. 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:

  8. 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).

  9. Case ..1.. Area Under a Curve

  10. Example ..1.. The graph below shows the curve and is shaded in the region

  11. The area is found by integrating

  12. Example ..2..

  13. Case ..2.. Area between two curves

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

  15. Area under f(x)= • Area under g(x)=

  16. Superimposing the two graphs: Area bounded by f(x) and g(x)

  17. Example ..3.. •  Find the area between the curves        y = 0      and      y = 3(x3 - x)

  18. Example ..4.. • Find the area bounded by the curves y = x2 - 4x – 5 and y = x + 1

  19. Solving the equations simultaneously,           x + 1 = x2 - 4x - 5            x = -1 or x = 6 Required Area =

  20. Volume Of A Revolution

  21. 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:

  22. Rotation about y-axis:

  23. Example ..5.. • The volume that we are looking for is shown in the diagram below

  24. To find the volume, we integrate

  25. Thank u 4 listening

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