integration application (aplikasi integral)
TRANSCRIPT
Integration Application
THP-FTP-UB
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 = aand x = b:
where
( ) ( )
( ) ( )
bb
x a
x a
f x dx F x
F b F a
( ) ( )f x F x
Basic applications
Areas under curves
If the integral is negative then the area lies below the x-axis. For example:
33 32 2
1 1
13
13
( 6 5) 3 53
( 3) (2 )
5
x x
xx x dx x x
In order to make it easy,you may sketch the function graph first
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 𝒚𝟐 = 𝟒𝒙 and
4x – 3y = 4
The Area Bounded By the Curves in Form of
Parametric Equations
Example
Try!
Find the area of the indicated region
A curve has parametric equations 𝑥 = 𝑐𝑜𝑠2𝑡, 𝑦 = 3𝑠𝑖𝑛2𝑡. Find thearea bounded by the curve, the x-axis and the ordinates at 𝑡 =0 𝑎𝑛𝑑 𝑡 = 2𝜋
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:
We finally obtain:
Volumes of Solid of Revolutions:Method of Washers
Sometimes, slicing a solid of revolution result in disks with hole inthe middle
Find the volume of the solid generated by revolving the regionbounded by the parabolas 𝑦 = 𝑥2 and 𝑦2 = 8𝑥 about the 𝑥 −axis
15
Jika V(t) adl volume air dlm waduk pada waktu t, maka turunan V’(t) adllaju mengalirnya air ke dalam waduk pada waktu t.
)V(t)V(t dt (t)V' 12
2t
1t
perubahan banyaknya air dalam waduk diantara t1 dan t2
Penerapan Integral dalam Ilmu Sains
2t
1t
dtdt
d[C][C](t2)-[C](t1)
Jika [C](t) adl konsentrasi hasil suatu reaksi kimiapd waktu t,maka laju reaksi adl turunan d[C]/dt
perubahan konsentrasi C dari waktu t1 ke t2
17
Jika laju pertumbuhan populasi adl dn/dt, maka
)n(t)n(t dt dt
dn12
2t
1t
pertambahan populasi selama periode waktu t1 ke t2