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