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1 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
2 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
3 Hence (, d (, (, d (, Emple 3 3 Determine the ntiderivtive o with respect to d 3 C ( Double Integrtion Pge 3
4 Emple Find the ntiderivtive o with respect to. Double Integrtion Pge 4
5 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
6 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
7 Emple 4 Evlute the iterted integrl ( dd. 0 Double Integrtion Pge 7
8 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
9 Importnt result: 0 ( d d nd ( d d 0 Hve the SAME vlue!! Double Integrtion Pge 9
10 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
11 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
12 = ( (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
13 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
14 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
15 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
16 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
17 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
18 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
19 Emple 6 Evlute the double integrl bounded b the curves = nd = 3. da, where is the region = (, = 3 Double Integrtion Pge 9
20 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
21 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
22 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
23 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
24 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
25 Assume the surce densit o the mteril is. Mss, M = Centrod, = (, dd = dd 0 0 d 0 0 = M (, dd 3 dd 4 d Moment o inerti = = (, dd 3 4 dd d Double Integrtion Pge 5
26 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
Multiple Integrals. Review of Single Integrals. Planar Area. Volume of Solid of Revolution
Multiple Integrls eview of Single Integrls eding Trim 7.1 eview Appliction of Integrls: Are 7. eview Appliction of Integrls: olumes 7.3 eview Appliction of Integrls: Lengths of Curves Assignment web pge
Multiple Integrals. Review of Single Integrals. Planar Area. Volume of Solid of Revolution
Multiple Integrls eview of Single Integrls eding Trim 7.1 eview Appliction of Integrls: Are 7. eview Appliction of Integrls: Volumes 7.3 eview Appliction of Integrls: Lengths of Curves Assignment web pge
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