1. smaller circle with the center at (0,

1.     The interpretation of the problem is shown in the figure below.

The total area for the goat can move around is equal to the sum of the area shape shaded in yellow and the area of the semi-circle shaded in blue. By geometry the area of the semi-circle can be directly of obtain using the formula . The problem here is the determination of the area shaded in yellow which is an irregular area where there is no direct formula that can be used. The idea here is to solve for the area using integration.

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f(x)

x

dx

y

y2

y1

x1

x2

(0, 0)

(2.5, 0)

(5.5, 0)

Using integration we need to find the function/s is or are involve in the calculation. Adding the Cartesian coordinates system in the figure above will help us obtain the required functions.

The necessary functions are y1 and y2 both as a function of x. With the use of the standard equation of the circle,  where (h, k) is center of the circle and r as the radius, the functions can be obtained. The smaller circle with the center at (0, 0) has an equation express as, . Considering only the function above the x axis, thus . The larger circle with center at (2.5, 0) will have an equation, . Therefore the function above the x-axis is taken as .

We need also to consider the limits for the bounded area in x direction. The shaded area is bounded by the vertical at x2 = 2.5 and the point of intersection of the two functions, say at x1. The point of intersections can considering only the functions above the x-axis can be calculated by equating the two functions. The calculation of the point of intersection is shown below.

(1)                                                   equate the two functions

(2)                   substitution

(3)                         squaring both sides

(4)                expansion of the binominal

(5)                 distribution

(6)                                                   solve for x

Therefore the point of intersection of the two functions is at x = 7/10. Since the functions are both above the x axis, so we can solve for the area above the x-axis and afterwards double it. The required area shaded in yellow can now be determine by evaluating , where y is the height of the vertical strip and at any point equal to the difference of y2 and y1. Thus the integral now becomes . Using numerical methods, the integration is carried and shown below in tabular form.

Evaluate:

 

 

Upper limit       (b)

Lower Limit   (a)

Number of subdivisions (n)

Function:

 

 

 

2.5

0.7

30

Width per strip (h)

0.060000000000

i

xi

f(xi)

Area per Strip by Trapezoidal Rule

Area per two Strips by Simpson’s Rule

Difference of Areas

0

0.70

0.00000000

 

 

 

1

0.76

0.06216934

0.0018650803

 

 

2

0.82

0.12378351

0.0055785857

0.0074492177

-0.000005552

3

0.88

0.18499505

0.0092633569

 

 

4

0.94

0.24594962

0.0129283402

0.0221942667

-0.000002570

5

1.00

0.30678836

0.0165821396

 

 

6

1.06

0.36764999

0.0202331507

0.0368150614

0.000000229

7

1.12

0.42867285

0.0238896854

 

 

8

1.18

0.48999695

0.0275600941

0.0514467672

0.000003012

9

1.24

0.55176601

0.0312528888

 

 

10

1.30

0.61412977

0.0349768733

0.0662238152

0.000005947

11

1.36

0.67724642

0.0387412856

 

 

12

1.42

0.74128553

0.0425559583

0.0812880193

0.000009225

13

1.48

0.80643140

0.0464315078

 

 

14

1.54

0.87288725

0.0503795594

0.0967979674

0.000013100

15

1.60

0.94088033

0.0544130274

 

 

16

1.66

1.01066857

0.0585464671

0.1129415430

0.000017952

17

1.72

1.08254913

0.0627965311

 

 

18

1.78

1.15687000

0.0671825741

0.1299547021

0.000024403

19

1.84

1.23404578

0.0717274734

 

 

20

1.90

1.31458001

0.0764587736

0.1481526623

0.000033585

21

1.96

1.39909777

0.0814103335

 

 

22

2.02

1.48839483

0.0866247781

0.1679873186

0.000047793

23

2.08

1.58351543

0.0921573080

 

 

24

2.14

1.68588212

0.0980819267

0.1901667739

0.000072461

25

2.20

1.79752810

0.1045023067

 

 

26

2.26

1.92155345

0.1115724467

0.2159509597

0.000123794

27

2.32

2.06314465

0.1195409431

 

 

28

2.38

2.23235398

0.1288649589

0.2481297206

0.000276181

29

2.44

2.45497364

0.1406198285

 

 

30

2.50

3.00000000

0.1636492091

0.3010449705

0.003224067

 

 

 

 

 

 

 

 

 

 

 

 

Sum of the Areas using areas per strip

1.880387392

1.876543766

0.003843626

Total area using the formula

1.880387392

1.876543766

0.003843626

 

The total area for the goat to move around is shown in the table below:

AREAS

TRAPEZOIDAL RULE

SIMPSON’S RULE

Area of the yellow shaded part above the x-axis

1.880387392 m2

1.876543766 m2

Total Area of the yellow shaded part

3.760774784 m2

3.753087532 m2

Area of the semi-circle

14.1372 m2

Total Area

17.89794173 m2

17.89025447 m2

 

2.    

f(x)

dx

y

y2

y1

x1

x2

(0, 4)

cg

 

y

The graph of the parabola, and the line is shown below. The volume of the geometric shape about an axis can determine using 2nd Pappus Theorem. According to Pappus the volume to be generated of a plane figure revolved about an axis is equal to distance traveled by the centroid of the plane figure multiplied by its area.

 

Considering the differential strip shown on the right, the differential volume generated is express as, . Then apply integration to solve for the volume generated by revolving the area about the x-axis. For the plane area, the points of intersection are once again determined. The calculation are as follows:

(1)                                                  equate the two functions

(2)                             substitution

(3)                            solve for x

Thus, the limits of the plane area are x1 = -2 and x2 =1.

 

The location of the center of gravity of the vertical strip is equal to  where at any point is equal to the average of y1 and y2. The area of the vertical strip is equal to the difference of y1 and y2 multiplied by the differential distance dx. The volume generated can be simplified as shown below.

(1)

(2)

(3)

Using numerical techniques, the integration is employed and shown below in tabular form.

Evaluate:

 

 

 

Upper limit       (b)

Lower Limit   (a)

Number of subdivisions (n)

Function:

 

1

-2

30

Width per strip (h)

0.100000000000

i

xi

f(xi)

Area per Strip by Trapezoidal Rule

Area per two Strip by Simpson’s Rule

Difference of Areas

0

-2.00

0.00000000

 

 

 

1

-1.90

0.44642032

0.0223210158

 

 

2

-1.80

1.68892021

0.1067670263

0.1158200492

0.013267993

3

-1.70

3.58801297

0.2638466590

 

 

4

-1.60

6.01175170

0.4799882336

0.7350907930

0.008744100

5

-1.50

8.83572934

0.7423740520

 

 

6

-1.40

11.94307863

1.0389403985

1.7765915896

0.004722861

7

-1.30

15.22447216

1.3583775395

 

 

8

-1.20

18.57812232

1.6901297237

3.0473029861

0.001204277

9

-1.10

21.90978133

2.0243951821

 

 

10

-1.00

25.13274123

2.3521261277

4.3783329616

-0.001811652

11

-0.90

28.16783389

2.6650287560

 

 

12

-0.80

30.94343100

2.9555632446

5.6249169265

-0.004324926

13

-0.70

33.39544407

3.2169437534

 

 

14

-0.60

35.46732442

3.4431384244

6.6664177230

-0.006335545

15

-0.50

37.11006322

3.6288693821

 

 

16

-0.40

38.28219144

3.7696127330

7.4063256248

-0.007843510

17

-0.30

38.94977988

3.8615985659

 

 

18

-0.20

39.08643916

3.9018109519

7.7722583371

-0.008848819

19

-0.10

38.67331972

3.8879879442

 

 

20

0.00

37.69911184

3.8186215784

7.7159609967

-0.009351474

21

0.10

36.16004560

3.6929578723

 

 

22

0.20

34.05989091

3.5109968258

7.2133061722

-0.009351474

23

0.30

31.40995751

3.2734924212

 

 

24

0.40

28.22909495

2.9819526229

6.2642938634

-0.008848819

25

0.50

24.54369261

2.6386393777

 

 

26

0.60

20.38767968

2.2465686145

4.8930515019

-0.007843510

27

0.70

15.80252521

1.8095102446

 

 

28

0.80

10.83723802

1.3319881612

3.1478339510

-0.006335545

29

0.90

5.54836679

0.8192802402

 

 

30

1.00

0.00000000

0.2774183393

1.1010235053

-0.004324926

Sum of the Volumes using volume per strip

67.81124601

67.85852698

-0.047280969

Total volume using the formula

67.81124601

67.85852698

-0.04728096

 

3.     The solution to problem 3 is summarized as follows:

(1)                                                              x in terms of t

(2)                                                         derivative of x with respect to t

(3)                                                           y in terms of t

(4)                                                              derivative of y with respect to t

(5)                                            derivative of y with respect to x

(6)  

(7)                                          the length of curve

(8)   At t = 0, x = 0 and at t =1, x = 1                                    determining the limits of the integral

(9)                                            the integral expression for the problem

The evaluation of the integral using trapezoidal and Simpson’s rule is shown by the table below.

Evaluate:

 

 

 

Upper limit       (b)

Lower Limit   (a)

Number of subdivisions (n)

Function:

 

 

 

1.0000000

0.0000000

30

Width per strip (h)

0.0333333

i

xi

f(xi)

Area per Strip by Trapezoidal Rule

Area per Strip by Simpson’s Rule

Difference of Areas

0.0000000

0.0000000

1.0000000

 

 

 

1.0000000

0.0333333

1.0198039

0.0336634

 

 

2.0000000

0.0666667

1.0770330

0.0349473

0.0684028

0.0002079

3.0000000

0.1000000

1.1661904

0.0373871

 

 

4.0000000

0.1333333

1.2806248

0.0407803

0.0780269

0.0001404

5.0000000

0.1666667

1.4142136

0.0449140

 

 

6.0000000

0.2000000

1.5620499

0.0496044

0.0944392

0.0000792

7.0000000

0.2333333

1.7204651

0.0547086

 

 

8.0000000

0.2666667

1.8867962

0.0601210

0.1147856

0.0000440

9.0000000

0.3000000

2.0591260

0.0657654

 

 

10.0000000

0.3333333

2.2360680

0.0715866

0.1373263

0.0000256

11.0000000

0.3666667

2.4166092

0.0775446

 

 

12.0000000

0.4000000

2.6000000

0.0836102

0.1611389

0.0000158

13.0000000

0.4333333

2.7856777

0.0897613

 

 

14.0000000

0.4666667

2.9732137

0.0959815

0.1857325

0.0000103

15.0000000

0.5000000

3.1622777

0.1022582

 

 

16.0000000

0.5333333

3.3526109

0.1085815

0.2108326

0.0000071

17.0000000

0.5666667

3.5440090

0.1149437

 

 

18.0000000

0.6000000

3.7363083

0.1213386

0.2362773

0.0000050

19.0000000

0.6333333

3.9293765

0.1277614

 

 

20.0000000

0.6666667

4.1231056

0.1342080

0.2619658

0.0000037

21.0000000

0.7000000

4.3174066

0.1406752

 

 

22.0000000

0.7333333

4.5122057

0.1471602

0.2878326

0.0000028

23.0000000

0.7666667

4.7074409

0.1536608

 

 

24.0000000

0.8000000

4.9030603

0.1601750

0.3138337

0.0000021

25.0000000

0.8333333

5.0990195

0.1667013

 

 

26.0000000

0.8666667

5.2952809

0.1732383

0.3399380

0.0000017

27.0000000

0.9000000

5.4918121

0.1797849

 

 

28.0000000

0.9333333

5.6885851

0.1863400

0.3661235

0.0000013

29.0000000

0.9666667

5.8855756

0.1929027

 

 

30.0000000

1.0000000

6.0827625

0.1994723

0.3923739

0.0000011

 

 

 

 

 

 

 

 

 

 

 

 

Sum of the length using length per strip

3.2495776

3.2490296

0.0005480

Total length of curve using the formula

3.2495776

3.2490296

0.0005480