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Application of Integral

Application of Integral. Done by : Fatma Al- nuaimi Kashaf Bakali Parisa Yazdjerdi Heba Hammud Nadine Bleibel. Definition of Area by Integral by : Fatma Al- nuaimi (201004421). Finding areas by integration. Using Riemann sum. Use Riemann sum to find the value of:. The graph:.

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Application of Integral

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  1. Application of Integral Done by : Fatma Al-nuaimi KashafBakali Parisa Yazdjerdi HebaHammud Nadine Bleibel

  2. Definition of Area by Integral by : Fatma Al-nuaimi(201004421)

  3. Finding areas by integration. Using Riemann sum.

  4. Use Riemann sum to find the value of: The graph:

  5. Step 1: • We can determine the value by subdividing the region into rectangle: • When the number of rectanglesn • The area of rectangle is A=L*W • Width=W= 1/n • Length= 1+(1/n)i • So,

  6. Step 3: • Performing some algebraic manipulation:

  7. Step 4: • Taking the limit to calculate the area:

  8. Area under the curveBy: Kashaf Bakali (201105803)

  9. Find the Area enclosed by the parabola and above the x-axis. • As the area to be calculated should be above x-axis so, . • We first find the points of intersection by solving both equations simultaneously. i.e. • So, Hence, the intersection points are; .

  10. Now, we sketch the graph. • The graph tells us the limit. • In this case, we have to find area from -1 till 3. • So, dx ) =

  11. Similarly,

  12. Find the area bounded between and • Firstly, we would find the intersection points by solving both given equations simultaneously. i.e. • We get the intersections points as, • We sketch the graph.

  13. Now we have, . • We use the formula, • In this case, and. • Hence,

  14. Area between two curveBy: Parisa Yazdjerdi (201005599)

  15. Areas between two curves • Process of finding area between two curves consist of 3 main steps : • finding intersection of the curves ( put two equation in an equality) • Drawing the graph to distinguish intervals and exact areas • Using integral formula

  16. Area between two curves • Example : find area between f(x) = Sin x , g(x) = Cos x , x = 0 and x = π/2 . • First step : find intersections

  17. Area between tow curves • Second step : Drawing the Graph

  18. Area between two curves • Third step : Using formula to find area

  19. Area between two curves

  20. Volume of SolidBy : HebbaHammud(201003247)

  21. Volume of Solids • For any Solid(S),we cut it into pieces and approximate each piece by a cylinder. This is called : cross-sectional area.

  22. Definition of Volume

  23. Exercises(about the x-axis)

  24. Find the volume v resulting from the revolution of the region bounded by:y=√x , from x=0 to x=1 about the x-axis.

  25. Find the volume v resulting from the revolution of the region bounded by:y=√(a2-x2 ) from x=-a to x=a and the x-axis about the x-axis.

  26. Exercises (about the y-axis)

  27. Find the volume of the solid of revolution generated by rotating the curve y = x3 between y = 0 and y = 4 about the y-axis. We first must express x in terms of y, so that we can apply the formula. If y = x3 then x = y1/3 The formula requires x2, so x2 = y2/3

  28. Find the volume generated by the areas bounded by the Given curves if they are revolved about the y-axis:y2 = x, y = 4 and x = 0 [revolved about the y-axis]

  29. VolumeBy: Nadine Bleibel(201104593)

  30. Basics of a Cylinder A cylinder is a simple solid which is bounded by a plane region B1- which is called the base. A cylinder also has a congruent region B2 in a Parallel plane. The formula for volume for a circular cylinder is V=(Pi)r^2(h)

  31. EXAMPLE 1 (example 2 p 356) Find the Volume of the solid obtained by rotating About the x-axis the region under the curve y= √x from 0 to 1.

  32. EXAMPLE 2 (example 3 p357) EXAMPLE: Find the volume of the solid obtaining by rotating about the y-axis the region bounded by y = x3, y = 8, and x = 0.

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