258

UNIT

4.1 MULTIPLE INTEGRALS

4.2 DOUBLE INTEGRALS

(x , y )r r

O x = a x = b

yr

xrR

y = f (x)1

y = f (x)2

Y

X

MULTIPLE INTEGRALS 259

4.3 WORKING RULE

4.4 DOUBLE INTEGRATION FOR POLAR CURVES

O

K2K1

L1L2 AP (r, )

Q (r + r, + )

B

260 A TEXTBOOK OF ENGINEERING MATHEMATICS—I

MULTIPLE INTEGRALS 261

O X

Y

262 A TEXTBOOK OF ENGINEERING MATHEMATICS—I

O Xx = 1y = x2

A

Y

MULTIPLE INTEGRALS 263

O(4, 0)G E

B(8, 2)xy = 16

X

A (4, 4)

x = 8

y = xY

O y = 0 Xx = a

x = 0y = a – x2 2

Y

264 A TEXTBOOK OF ENGINEERING MATHEMATICS—I

O (0, 0) X

y = x

(4, 4)B

Y

Y(0, b)

X(a, 0)

MULTIPLE INTEGRALS 265

O = = 0

A X

P (r, )Q = –2

Y

266 A TEXTBOOK OF ENGINEERING MATHEMATICS—I

EXERCISE 4.1

MULTIPLE INTEGRALS 267

268 A TEXTBOOK OF ENGINEERING MATHEMATICS—I

4.5 CHANGE OF THE ORDER OF INTEGRATION

MULTIPLE INTEGRALS 269Y

x

x XO

O x = a

L A (a, a)

x = ay2

x + y = 2aB (0, 2a)

270 A TEXTBOOK OF ENGINEERING MATHEMATICS—I

O x = 2 X

B y = 1(0, 1)A

C (2, e )2

Y

A

Gx = –2B (0,0) X

x = 1

F (0, 2)y = x + 4x2

y = 3x + 2E (1, 5)

Y

MULTIPLE INTEGRALS 271

X Ox = 0

C

x = 1 X

B (1, 1)

x = 0

A(0, 2)

Y

272 A TEXTBOOK OF ENGINEERING MATHEMATICS—I

(2a, 0) O

B

A

x = 0 x = a

N (a, a)

(0, 2a)P

M(a, 3a)

x = y – 2a

K

O y = 0 X

x = 0 x = y—a2

Ay = aYB

MULTIPLE INTEGRALS 273

Ox = 4ay2

X

(4a, 4a)A

Y

274 A TEXTBOOK OF ENGINEERING MATHEMATICS—I

4.6 CHANGE OF VARIABLES IN A MULTIPLE INTEGRAL

O(0,0)

Y

Q

X

MULTIPLE INTEGRALS 275

O X

Y

P

Qv + v

u + uQ

P

v

u

276 A TEXTBOOK OF ENGINEERING MATHEMATICS—I

MULTIPLE INTEGRALS 277

RO

Y

X R*

v = 3

v

u

u = 2O

278 A TEXTBOOK OF ENGINEERING MATHEMATICS—I

O r = 0 = 0 X

r

Y

O (0, 0) A (1, 0) X

R x = 1

B (1, 1)Y

MULTIPLE INTEGRALS 279

vA (1, 1) v = 1 B (4, 1)

O u

u = 4u = 1

V = –2 C (4, –2)D(1, –2)

280 A TEXTBOOK OF ENGINEERING MATHEMATICS—I

x=0

B (0,1)

O

x +y =12 2

A(1,0)

MULTIPLE INTEGRALS 281

EXERCISE 4.2

282 A TEXTBOOK OF ENGINEERING MATHEMATICS—I

MULTIPLE INTEGRALS 283

4.7 BETA AND GAMMA FUNCTIONS

4.7.1 Properties of Beta and Gamma Functions

284 A TEXTBOOK OF ENGINEERING MATHEMATICS—I

MULTIPLE INTEGRALS 285

4.8 TRANSFORMATIONS OF GAMMA FUNCTION

286 A TEXTBOOK OF ENGINEERING MATHEMATICS—I

4.9 TRANSFORMATIONS OF BETA FUNCTION

MULTIPLE INTEGRALS 287

4.10 RELATION BETWEEN BETA AND GAMMA FUNCTIONS

4.11 SOME IMPORTANT DEDUCTIONS

288 A TEXTBOOK OF ENGINEERING MATHEMATICS—I

MULTIPLE INTEGRALS 289

4.12 DUPLICATION FORMULA

290 A TEXTBOOK OF ENGINEERING MATHEMATICS—I

MULTIPLE INTEGRALS 291

292 A TEXTBOOK OF ENGINEERING MATHEMATICS—I

MULTIPLE INTEGRALS 293

294 A TEXTBOOK OF ENGINEERING MATHEMATICS—I

4.13 EVALUATE THE INTEGRALS

MULTIPLE INTEGRALS 295

296 A TEXTBOOK OF ENGINEERING MATHEMATICS—I

MULTIPLE INTEGRALS 297

298 A TEXTBOOK OF ENGINEERING MATHEMATICS—I

MULTIPLE INTEGRALS 299

EXERCISE 4.3

300 A TEXTBOOK OF ENGINEERING MATHEMATICS—I

4.14 APPLICATION TO AREA (DOUBLE INTEGRALS)4.14.1 Area in Cartesian Coordinates

MULTIPLE INTEGRALS 301

O X

Y

y = aA B

PQ

NM x

y

Cx = f(y)

D y = b

O X

Y

M NA

By = f (x)2

y x = b

CD

x = a

y = f (x)1

x

O B N M C X

A

Y

x = ax

y x = b

(x, y) P QD

302 A TEXTBOOK OF ENGINEERING MATHEMATICS—I

O X

Y

(0, 0) CD (4, 0)

x y A (3, 3)B

X a O a

Y

(a, 0) XAP(x,y)C

Y

MULTIPLE INTEGRALS 303

O X

Y

(0, 1) (2, 1)xy = 2

(0, 4) y = 4

(2, 0)

O X

By

A (0, 2a)

Y

304 A TEXTBOOK OF ENGINEERING MATHEMATICS—I

4.14.2 Area of Curves in Polar Coordinate

Y

X A O N(4a, 0) A (8a, 0) X

P

Y

(8a, 0)

P

O

C

DE

F A =

P (r, )

B =

X

MULTIPLE INTEGRALS 305

O (0, 0)

A3

2 2( , )2 3

4

( )24

X

YB (0,a)

O (A,0)A

x + y = a

306 A TEXTBOOK OF ENGINEERING MATHEMATICS—I

Y

CircleCardioid

XOR

MULTIPLE INTEGRALS 307A

BO D

= /2

r = a

=

C

= 0

O = 0 X

r = a sin r = a (1 – cos )

—2 =

OD

C

A = 0

B

= /2F

308 A TEXTBOOK OF ENGINEERING MATHEMATICS—I

Y

OX

= 4

= 4

r = a cos2

MULTIPLE INTEGRALS 309

(a,o)

2

2

(2 a,o)

310 A TEXTBOOK OF ENGINEERING MATHEMATICS—I

Y

OX

x = a

NxM

A X(a, 0)

PQ

Y B

X O Nx

M

PQ

Y

A (2a, 0) X

x = 2a

Y B

MULTIPLE INTEGRALS 311

EXERCISE 4.4

312 A TEXTBOOK OF ENGINEERING MATHEMATICS—I

4.15 TRIPLE INTEGRALS

MULTIPLE INTEGRALS 313

Z z = a

OS

z = 0

X

Y

314 A TEXTBOOK OF ENGINEERING MATHEMATICS—I

EXERCISE 4.5

Y(0, 2, 0)

z = 0 dxdy

O X(2, 0, 0)

z = 4–x2(0, 0, 4)

Z

MULTIPLE INTEGRALS 315

4.16 APPLICATION TO VOLUME (TRIPLE INTEGRALS)

316 A TEXTBOOK OF ENGINEERING MATHEMATICS—I

Z

z = x + 9y2 2

OD

X(3, 0, 0) x + 9y = 92 2Y(0,1,0)

MULTIPLE INTEGRALS 317

318 A TEXTBOOK OF ENGINEERING MATHEMATICS—I

x = y2

O X

y = x2

Y

X

Z

O

Y

(a, 0)x + y = ax2 2

MULTIPLE INTEGRALS 319

Z

x + y + z = a2 2 2 2

Y

X

O

x + y = z2 2 2

320 A TEXTBOOK OF ENGINEERING MATHEMATICS—I

MULTIPLE INTEGRALS 321

az=x +y2 2

x + y = a2 2 2

322 A TEXTBOOK OF ENGINEERING MATHEMATICS—I

EXERCISE 4.6

MULTIPLE INTEGRALS 323

4.17 DRITCHLET’S* THEOREM

YB

z = 0O A X

y = x y z

z = 1–x–yC

Z

324 A TEXTBOOK OF ENGINEERING MATHEMATICS—I

MULTIPLE INTEGRALS 325

326 A TEXTBOOK OF ENGINEERING MATHEMATICS—I

MULTIPLE INTEGRALS 327

328 A TEXTBOOK OF ENGINEERING MATHEMATICS—I

x

y

A

O

z(0,0,1)

(0,1,0)

(1,0,0)

B

MULTIPLE INTEGRALS 329

EXERCISE 4.7

330 A TEXTBOOK OF ENGINEERING MATHEMATICS—I

OBJECTIVE TYPE QUESTIONS

MULTIPLE INTEGRALS 331

332 A TEXTBOOK OF ENGINEERING MATHEMATICS—I

ANSWERS TO OBJECTIVE TYPE QUESTIONS