kuliah kalkulus 2 tahun 2013

8/18/2019 KULIAH Kalkulus 2 Tahun 2013

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CALCULUS 2

BY:

GUNARTO,M.EngDepartment of Mechanical Engineering

Muhammadiyah University of Pontianak


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CALCULUS 2

Referensi :

1.Paul A.Foerster, Calculus Concepts and Applications , Key Curriculum Press, 2005

2.Jerrold Marsden &Alan Weinstein, CalculusI,II,III , Springer Verlag New York Inc.,1985

3.Paul Dawkin, Calculus II , 2007

4.John Bird ,Higher Engineering Mathematics ,Fifth Edition, Published by Elsevier Ltd.,2006

5.Benjamin Crowell,Calculus, Light and Matter,2010

6.K.A.Stroud, Engineering Mathematics ,

Industrial Press,Inc. New york, 2001


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Find;


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Find;


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Find;


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Find;


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Contoh soal:

1.Tentukanlah integral berikut :

2.a.Tentukanlah I = diketahui bahwa I =25

apabila x =3b.Tentukanlah I = diketahui bahwa I =16

apabila x = 2

dxxcos2h. sin5.

dxxg. 6

.

4 j. dx8f. 3.

xi. sece. .

1/2

x

-326

xdxd

dx x

c

dxdxeb

dx xdxdx xa

x

dx x24

dx5


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1.

2.

3.

4.

5.

6.

Homework!


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HOMEWORK!

FIND:

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C


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k


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Homework!


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+C


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+C


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a x


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Homework!


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Homework!

Homework!


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Homework!

Homework!


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Homework!

Homework!


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Homework!

INTEGRASI DENGAN PECAHAN PARSIAL


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INTEGRASI DENGAN PECAHAN PARSIAL

Pernyataanseperti tidak

muncul dalam daftar integral standarkita,tetapi sebenarnya muncul pada banyakpenerapan matematis.

Sebenarnya,

C x

dx

x

dx

x

dx

x x

xdx

x x

x

sehingga x x x x

x x x

x

)12ln(2

1 5)(xln3

12

1

5

3

)12)(5(

87

5112

87

121

53

)12)(5(87

511287

2

2

dx x x

x

5112

872

18/03/2013 Gunarto

Aturan-aturan dari pecahan parsial adalah sbb:


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Aturan-aturan dari pecahan parsial adalah sbb:

a.Pembilang dari fungsi yang diberikan harusmemiliki derajat yang lebih rendah daripadapenyebutnya.Jika tidak demikian, makapertama-tama bagilah dengan menggunakanpembagian panjang.

b.Faktorkan penyebutnya menjadi faktor-faktor prima.Ini penting karena faktor-faktoryang diperoleh akan menentukan bentuk daripecahan parsial.

18/03/2013 Gunarto

c Faktor linear (ax + b) menjadi pecahan


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c.Faktor linear (ax + b) menjadi pecahanparsial

d.Faktor linear (ax + b)2 menjadi pecahanparsial

e.Faktor linear (ax + b)3 menjadi pecahanparsial

2)( bax

B

bax

A

18/03/2013 Gunarto

bax

A

32 )()( baxC

bax B

bax A

f Faktor kuadratik (ax2 + bx + c) menjadi


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f. Faktor kuadratik (ax + bx + c) menjadipecahan parsial

cbxax B Ax

2

18/03/2013 Gunarto


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Homework!


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Homework!


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Determine:


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Above

Homework!


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below


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figure


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Figure

Figure


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the figure


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the figure


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figure


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Homework!

D t i


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Determine:

The velocity v of a body t seconds after a certaininstant is: (2t2 + 5) m/s. Find by integration how

far it moves in the interval from t=0 to t=4 s

Sketch the graph y = x3 +2x2 - 5x - 6 between x=-3

and x=2 and determine the area enclosed by the

curve and the x-axis

Determine the area enclosed between the

curves y = x2 +1 and y=7-x

(a) Determine the coordinates of the points of

intersection of the curves y = x2 and y2 = 8x.

(b) Sketch the curves y = x2 and y2 = 8x on the same

axes.

(c) Calculate the area enclosed by the two curves

Homework!


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The curve y=x2+4 is rotated one revolutionabout the x-axis between the limits x=1 and

x=4. Determine the volume of the solid of

revolution produced

Two parabolas y=x2 and y2=8x is rotated

360° about the x-axis and y-axis. Determinethe volume of the solid produced

Centroids of simple shapes


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A lamina is a thin flat sheet having uniform thickness.

The centre of gravity of a lamina is the point where it balances perfectly, i.e. the lamina’s centre of mass.

When dealing with an area (i.e. a lamina of negligible

thickness and mass) the term centre of area or centroid

is used for the point where the centre of gravity of a

lamina of that shape would lie.

Centroids

The first moment of area


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The first moment of area is defined as the product of the area

and the perpendicular distance of its centroid from a given axis inthe plane of the area. In Fig. above the first moment of area Aabout axis XX is given by (Ay) cubic units.

Centroid of area between a curveand the x -axis


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The Figure shows an area PQRS bounded by thecurve y=f(x), the x-axis and ordinates x=a andx=b. Let this area be divided into a large numberof strips, each of width δx.A typical strip is shown shaded drawn at point

(x,y) on f(x).The area of the strip is approximately rectangularand is given by yδx. The centroid, C, hascoordinates (x, y/2).

First moment of area of shaded strip about axis

Oy= yδxx=xyδx.


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Centroid of area between a curve and the y -axis


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If x and y are the distances of the centroid of area EFGH in the Figurefrom Oy and Ox respectively, then, by similar reasoning as above:


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Show, by integration, that the centroid of a rectangle

lies at the intersectionof the diagonals


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lies at the intersectionof the diagonals

Determine by integration the position of the centroid of the area

enclosed by the line y=4x, the x-axis and ordinates x=0 and x=3


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enclosed by the line y 4x, the x axis and ordinates x 0 and x 3

Homework!


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Determine the co-ordinates of the centroid of thearea lying between the curve y=5x-x2 and the x-axis

Locate the centroid of the area enclosed by the curvey=2x2, the y-axis and ordinates y=1 and y=4,correct to 3 decimal places

Locate the position of the centroid enclosed by thecurves y=x2 and y2=8x

Second moments of area and radius of gyration


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Second moment of area of regular sections


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The procedure to determine the second moment of area of regular sections about a given axis is (i) to find the secondmoment of area of a typical element and (ii) to sum all suchsecond moments of area by integrating between appropriatelimits.

b


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In Figure, axis GG passes through thecentroid C of area A. Axes DD and GG are inthe same plane, are parallel to each other

d di t d t Th ll l iD

Parallel axis theorem


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and distance d apart. The parallel axistheorem states:

Using the parallel axis theorem the secondmoment of area of a rectangle about an axisthrough the centroid may be determined. Inthe rectangle shown in Figure,

D


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12

83

2

3

3

3

32

2

3

2

2

32

2

2

2

2

22

bh I

hb yb I

y

bdy yb I

bdy ydA y I

h

h

h

h

h

h

h

h

dA y I x2

b

h

b

yh

w


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12

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3

44

0

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0

32

0

2

bh I

hh

h

b I

yhy

h

b I

dy yhyh

b I

dy yhh

b y I

yhh

bw

x

x

h

x

h

x

h

x

Determine the second moment of area and the radius of gyration about axes AA, BB and CC for the rectangle


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Homework!


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Determine the second moment of areaand radius of gyration for the semicircleshown in Figure about axis XX

Determine the second moment of areaand radius of gyration about axis QQ of

the triangle BCD shown in Figure

Find the second moment of area andthe radius of gyration about axis for therectangle shown in Figure.

Second moments of areas of composite areas


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Determine the second moment of area and the radius of gyration about axis XX for the I-section shown in Figure.


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Homework!


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For the sections shown in Figure, find the second momentof area and the radius of gyration about axis XX.

kuliah kalkulus 2 tahun 2013

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