In Mathematics for Construction, we learnt that

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III DOUBLE INTEGATION THE ANTIDEIVATIVE OF FUNCTIONS OF VAIABLES In Mthemtics or Construction, we lernt tht the indeinite integrl is the ntiderivtive o ( d ( Double Integrtion Pge

Hence d d ( d ( The ntiderivtive o (, with respect to is denoted b (, d (, And the ntiderivtive o with respect to is denoted b (, d Double Integrtion Pge

Hence (, d (, (, d (, Emple 3 3 Determine the ntiderivtive o with respect to. 3 3 3 d 3 C ( Double Integrtion Pge 3

Emple Find the ntiderivtive o with respect to. Double Integrtion Pge 4

DEFINITE DEFINITE INTEGAL OF FUNCTIONS OF INTEGAL OF FUNCTIONS OF VAIABLES VAIABLES F b F d b, (, (, ( d F where, (, ( Emple 3 Evlute the deinite integrl. 0 ( d Pge Double Integrtion 5

ITEATED INTEGALS Successive Integrls ( The deinite integrl I (, d is unction o, written s F(, which cn be integrted gin: ( F ( d Hence I d, d is obtined b two ( successive integrtions, is clled n iterted integrl. Double Integrtion Pge 6

Emple 4 Evlute the iterted integrl ( dd. 0 Double Integrtion Pge 7

Interchnging the Order o Integrtion Evlute the iterted integrl in Emple 4 in the opposite order, tht is, integrte with respect to irst nd then : 0 ( d d Wht is the nswer? Double Integrtion Pge 8

Importnt result: 0 ( d d nd ( d d 0 Hve the SAME vlue!! Double Integrtion Pge 9

Vrible Limits o Integrtion Sometimes the limits o integrtion m not be constnts. Emple 5 E l t th it t d i t l d d Evlute the iterted integrl. 0 Double Integrtion Pge 0

CONCEPTS OF DOUBLE INTEGALS Deinite integrl or unctions o Single Vrible In Mthemtics or Construction, we lernt the deinite integrl or unctions o single vrible. b ( d lim ( s i n provided d tht t the limit it eists. n i Double Integrtion Pge

= ( (s i i- s i i b = i - i- Wht is the mening to the vlue o the deinite integrl ( d b? Double Integrtion Pge

Deinite integrl or unctions o two vribles t j The double integrl o s i (, n over is deined b (, da lim lim ( si, t j n i provided tht the limit eists. m m j Double Integrtion Pge 3

Let z (, be the height bove the - plne t point (,in closed region. z z = (, da Cn ou give mening to the vlue o the double integrl (, da? Double Integrtion Pge 4

Some properties o double integrls. c (, da c (, da, where c is constnt. [ (, g(, ] da da (, g(, da 3. (, da (, da (, da, i the res nd do not overlp Double Integrtion Pge 5

DOUBLE INTEGATION OVE A BOUNDED IEGULA EGION The points in region enclosed b the curves g ( nd g (. d g ( c g ( L b Double Integrtion Pge 6

I is verticll simple region then b g ( da (, g ( (, b, g( g ( d d I is horizontll simple region then d g ( (, da c g ( c d, (, d d g ( g ( Hence (, da b g g ( ( (, d d d c g g ( ( (, d d Double Integrtion Pge 7

Finding the limits o integrtion o verticll simple region. A sketch Sketch the region o integrtion nd lbel the bounding curves.. The -limits o integrtion Imgine verticl line L cutting through in the direction o incresing. Mrk the -vlues where the verticl line L enters nd leves. These re the -limits o integrtion nd usull unctions o. 3. The -limits o integrtion Choose -limits tht include ll the verticl lines through. Double Integrtion Pge 8

Emple 6 Evlute the double integrl bounded b the curves = nd = 3. da, where is the region = (, = 3 Double Integrtion Pge 9

APPLICATIONS OF DOUBLE INTEGATION Volume & Are Double integrtion is deined s I (, da where is the region in - plne nd z (, is surce bove the region. The volume bounded b surce z nd the region is deined s V (, da B putting the (,, the re o the region is deined s A dd Double Integrtion Pge 0

Mss A thin sheet o mteril o uniorm thickness covers region in the - plne. Suppose the sheet hs vring densit (, (in kg/m t ech point (, in the region. The totl t mss M o the sheet tis given b M (, dd (i.e. b putting (, (, (Note tht the densit is deined s the mss per unit re. Double Integrtion Pge

First Moment & Centre o Grvit The irst moment o n element o n re bout n is in the -plne is given b the product o the mss o the element nd the perpendiculr distnce between the element nd the is. M (, da where A is the re o the region. & M (, da The centre o grvit o n re is hence deined s M & M (, da (, da M M M M Double Integrtion Pge

Second Moment or Moment o Inerti The second moment o n element o n re bout n is in the -plne is given b the product o the mss o the element nd the squre o the perpendiculr distnce between the element nd the is. I (, da & I (, da where A is the re o the region. Double Integrtion Pge 3

Emple 7 Find the mss, centroid (rom is nd the moment o inerti bout the -is o the tringulr sheet which hs uniorm thickness nd densit in the region bounded b the lines = nd = s shown below. 0 Double Integrtion Pge 4

Assume the surce densit o the mteril is. Mss, M = Centrod, = (, dd = dd 0 0 d 0 0 = M (, dd 3 dd 4 d 0 0 0 0 3 Moment o inerti = = (, dd 3 4 dd d 0 0 3 0 0 4 3 Double Integrtion Pge 5

Function n sin cos e sinh cosh Indeinite integrl n n c -cos + c sin + c e + c cosh + c sinh + c tn ( c sinh cosh ( c ( c Double Integrtion Pge 6

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