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- 1. Integration Application THP-FTP-UB
- 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 ο ο( ) ( ) ( ) ( ) b b x a x a f x dx F x F b F a ο½ ο½ ο½ ο½ ο ο² ( ) ( )f x F xο’ο½
- 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 ο½ ο½ ο© οΉ ο ο« ο½ ο ο«οͺ οΊ ο« ο» ο½ ο ο ο½ ο ο²
- 4. In order to make it easy,you may sketch the function graph first
- 5. The Area of The Region Between Two Curves Example Find the area of the region between the curves π¦ = π₯4 and 2π₯ β π₯2. Answer Sketch the functions graph first, then finding where the two curves intersect. Try! Find the area of the region between π π = ππ and 4x β 3y = 4
- 6. The Area Bounded By the Curves in Form of Parametric Equations Example
- 7. Try! Find the area of the indicated region A curve has parametric equations π₯ = πππ 2 π‘, π¦ = 3π ππ2 π‘. Find the area bounded by the curve, the x-axis and the ordinates at π‘ = 0 πππ π‘ = 2π
- 8. Volumes of Solid of Revolutions: Method of Disks Let π be the volume of the solid generated. Since the solid generated is a flat cylinder, so π is:
- 9. We finally obtain:
- 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 π¦ = π₯2 and π¦2 = 8π₯ about the π₯ βaxis
- 11. 15 Jika V(t) adl volume air dlm waduk pada waktu t, maka turunan Vβ(t) adl laju mengalirnya air ke dalam waduk pada waktu t. )V(t)V(tdt(t)V' 12 2t 1t οο½ο² perubahan banyaknya air dalam waduk diantara t1 dan t2 Penerapan Integral dalam Ilmu Sains
- 12. ο² ο½ 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. 17 Jika laju pertumbuhan populasi adl dn/dt, maka )n(t)n(tdt dt dn 12 2t 1t οο½ο² pertambahan populasi selama periode waktu t1 ke t2

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