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INTEGRAL OF CALCULUS

INTEGRAL OF CALCULUS. Volume of a solid of a revolution (Intro). Nikenasih B, M.Si Mathematics Educational Department Faculty of Mathematics and Natural Science State University of Yogyakarta. Contents. Problem and Aim Solution : A-R-L Method The Steps of Solution Examples Exercises.

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INTEGRAL OF CALCULUS

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  1. INTEGRAL OF CALCULUS Volume of a solid of a revolution (Intro) Nikenasih B, M.Si Mathematics Educational Department Faculty of Mathematics and Natural Science State University of Yogyakarta

  2. Contents • Problem and Aim • Solution : A-R-L Method • The Steps of Solution • Examples • Exercises

  3. Problem and Aim f function of x domain a ≤ x ≤ b y-axis f(x) x-axis x = a x = b What is the volume?

  4. Solution : A-R-L Method • What we Know The volume of a cylinder • What we want to know The volume of an arbitrary solid of a revolution • How we do it We approximate the solid with a certain number of cylinders

  5. We rotate region bounded by f(x) = c, x = a, x = b and x-axis 3600 about x-axis y-axis f(x) = c c x-axis x = a x = b We get cylinder, with radius c and height b – a.

  6. The steps of solution (1) • Approximate the region with n-rectangle, obtained by partitioning the interval [a,b] Here we get n-subinterval [x0 = a,x1] , [x1,x2] , ... , [xn-1,xn = b] • Select a point at each subinterval, suppose xi* for ith-subinterval Here we get n-rectangles where ith-rectangle has width xi – xi-1 and length f(xi*)

  7. The steps of solution (2) • By rotating the rectangle, we get n-cylinders where ith-cylinder has radius f(xi*) and height (xi – xi-1) • The volume of a solid of a revolution can be approximated by the sum volume of n-cylinders.

  8. f(x) x0 x1* x1 x2 . . . xn-1 xn

  9. Examples (1) • Approximate the volume of the solid obtained by rotating the region bounded by and the x-axis about the x-axis

  10. Answer (1)

  11. Examples (2) • Approximate the volume of the solid obtained by rotating the region bounded by and about the x-axis

  12. Answer (2)

  13. Exercises • Approximate the volume of the solid obtained by rotating the region bounded by and the x-axis about the x-axis

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