Integration application (Aplikasi Integral)
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The document discusses the calculation of areas under curves using integrals, including both basic applications and examples of finding areas between curves. It also covers the volumes of solids of revolution using the methods of disks and washers, alongside examples for each method. Additionally, the document touches on applications of integrals in fields such as science and population growth.
![
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](https://image.slidesharecdn.com/integrationapplication-150131020017-conversion-gate01/85/Integration-application-Aplikasi-Integral-16-320.jpg)
Integration application (Aplikasi Integral)
- 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 ( ) ( )f x F x
- 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
- 4. In order to make it easy,you may sketch the function graph first
- 6. The Area of The Region Between Two Curves Example Find the area of the region between the curves 𝑦 = 𝑥4 and 2𝑥 − 𝑥2. Answer Sketch the functions graph first, then finding where the two curves intersect. Try! Find the area of the region between 𝒚 𝟐 = 𝟒𝒙 and 4x – 3y = 4
- 7. The Area Bounded By the Curves in Form of Parametric Equations Example
- 8. Try! Find the area of the indicated region A curve has parametric equations 𝑥 = 𝑐𝑜𝑠2 𝑡, 𝑦 = 3𝑠𝑖𝑛2 𝑡. Find the area bounded by the curve, the x-axis and the ordinates at 𝑡 = 0 𝑎𝑛𝑑 𝑡 = 2𝜋
- 9. 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:
- 13. 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 = 𝑥2 and 𝑦2 = 8𝑥 about the 𝑥 −axis
- 15. 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
- 16. 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
- 17. 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