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Integration application (Aplikasi Integral)


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Materi kuliah tentang Aplikasi Integral. Cari lebih banyak mata kuliah Semester 1 di:

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Integration application (Aplikasi Integral)

  1. 1. Integration Application THP-FTP-UB
  2. 2. Basic applications Areas under curves The area above the x-axis between the values x = a and x = b and beneath the curve in the diagram is given as the value of the integral evaluated between the limits x = a and x = b: where  ( ) ( ) ( ) ( ) b b x a x a f x dx F x F b F a ο€½ ο€½ ο€½ ο€½ ο€­  ( ) ( )f x F xο‚’ο€½
  3. 3. Basic applications Areas under curves If the integral is negative then the area lies below the x-axis. For example: 33 3 2 2 1 1 1 3 1 3 ( 6 5) 3 5 3 ( 3) (2 ) 5 x x x x x dx x x ο€½ ο€½  οƒΉ ο€­  ο€½ ο€­ οƒͺ οƒΊ   ο€½ ο€­ ο€­ ο€½ ο€­ 
  4. 4. In order to make it easy,you may sketch the function graph first
  5. 5. The Area of The Region Between Two Curves Example Find the area of the region between the curves 𝑦 = π‘₯4 and 2π‘₯ βˆ’ π‘₯2. Answer Sketch the functions graph first, then finding where the two curves intersect. Try! Find the area of the region between π’š 𝟐 = πŸ’π’™ and 4x – 3y = 4
  6. 6. The Area Bounded By the Curves in Form of Parametric Equations Example
  7. 7. Try! Find the area of the indicated region A curve has parametric equations π‘₯ = π‘π‘œπ‘ 2 𝑑, 𝑦 = 3𝑠𝑖𝑛2 𝑑. Find the area bounded by the curve, the x-axis and the ordinates at 𝑑 = 0 π‘Žπ‘›π‘‘ 𝑑 = 2πœ‹
  8. 8. Volumes of Solid of Revolutions: Method of Disks Let 𝑉 be the volume of the solid generated. Since the solid generated is a flat cylinder, so 𝑉 is:
  9. 9. We finally obtain:
  10. 10. Volumes of Solid of Revolutions: Method of Washers Sometimes, slicing a solid of revolution result in disks with hole in the middle Find the volume of the solid generated by revolving the region bounded by the parabolas 𝑦 = π‘₯2 and 𝑦2 = 8π‘₯ about the π‘₯ βˆ’axis
  11. 11. 15 Jika V(t) adl volume air dlm waduk pada waktu t, maka turunan V’(t) adl laju mengalirnya air ke dalam waduk pada waktu t. )V(t)V(tdt(t)V' 12 2t 1t  perubahan banyaknya air dalam waduk diantara t1 dan t2 Penerapan Integral dalam Ilmu Sains
  12. 12.  ο€½ 2t 1t dt dt d[C] [C](t2)-[C](t1) Jika [C](t) adl konsentrasi hasil suatu reaksi kimia pd waktu t,maka laju reaksi adl turunan d[C]/dt perubahan konsentrasi C dari waktu t1 ke t2
  13. 13. 17 Jika laju pertumbuhan populasi adl dn/dt, maka )n(t)n(tdt dt dn 12 2t 1t  pertambahan populasi selama periode waktu t1 ke t2