One should study Mathematics because it is only through Mathematics that nature can be conceived in harmonious form. – Birkhoff 

8.1 Hkwfedk (Introduction)

T;kfefr esa] geus f=kHkqtksa vk;rksa] leyac prqHkqZtksa ,oa o`Ùkksa lfgr fofHkUu T;kferh; vko`Qfr;ksa osQ {ks=kiQy osQ ifjdyu osQ fy, lw=kksa dk vè;;u fd;k gSA okLrfod thou dh vusd leL;kvksa osQ fy, xf.kr osQ vuqiz;ksx esa bl izdkj osQ lw=k ewy gksrs gSaA izkjafHkd T;kfefr osQ lw=kksa dh lgk;rk ls ge vusd lk/kj.k vko`Qfr;ksa osQ {ks=kiQy dk ifjdyu dj ldrs gSaA ;|fi ;s lw=k oØksa }kjk f?kjs {ks=kiQy osQ ifjdyu osQ fy, vi;kZIr gSa blosQ fy, gesa lekdyu xf.kr dh oqQN ladYiukvksa dh vko';drk gksxhA

fiNys vè;k; esa geus ;ksxiQy dh lhek osQ :i esa fuf'pr lekdyuksa dk ifjdyu djrs le; oØ y = f (x), dksfV;ksa
x = a, x = b ,oa x-v{k ls f?kjs {ks=kiQy dks Kkr djus dk vè;;u fd;k gSA bl vè;k; esa ge lk/kj.k oØksa osQ varxZr] ljy js[kkvksa ,oa o`Ùkksa] ijoy;ksa] rFkk nh?ko`Ùkksa (osQoy ekud :i) dh pkiksa osQ chp f?kjs {ks=kiQy dks Kkr djus osQ fy, lekdyuksa osQ ,d fof'k"V vuqiz;ksx dk vè;;u djsaxsA mijksDr oØksa ls f?kjs {ks=kiQy dks Hkh Kkr djsaxsA

8.2 lk/kj.k oØksa osQ varxZr {ks=kiQy (Area Under Simple Curves)

fiNys vè;k; esa geus] ;ksxiQy dh lhek osQ :i esa fuf'pr lekdyu ,oa dyu dh vk/kjHkwr izes; dk mi;ksx djrs gq, fuf'pr lekdyu dk ifjdyu oSQls fd;k tk,] dk vè;;u fd;k gSA vc ge oØ y = f(x), x-v{k ,oa dksfV;k¡ x = a rFkk x = b ls f?kjs {ks=kiQy dks Kkr djus dh vklku ,oa varKkZu ls izkIr fof/ dh ppkZ djrs gSaA vko`Qfr 8-1 ls ge oØ osQ varxZr {ks=kiQy dks cgqr lh iryh ,oa mèokZ/j cgqr lh ifV~V;ksa ls fufeZr eku ldrs gSaA y m¡pkbZ ,oa dx pkSM+kbZ okyh ,d LosPN iV~Vh ij fopkj dhft,] blesa dA (izkjafHkd iV~Vh dk {ks=kiQy) = ydx, tgk¡ y = f(x) gSA

;g {ks=kiQy izkjafHkd {ks=kiQy dgykrk gS tks fd {ks=k osQ Hkhrj fdlh LosPN fLFkfr ij LFkkfir gS ,oa a rFkk b osQ eè; x osQ fdlh eku ls fofufnZ"V gSA oØ y = f (x), dksfV;ksa x = a, x = b ,oa x&v{k ls f?kjs {ks=k osQ oqQy {ks=kiQy A dks] {ks=k PQRSP esa lHkh iryh ifV~V;ksa osQ {ks=kiQyksa osQ ;ksxiQy osQ ifj.kke osQ :i esa ns[k ldrs gSaA lkaosQfrd Hkk"kk esa ge bls bl izdkj vfHkO;Dr djrs gSa%

A =

oØ x = g (y), y-v{k ,oa js[kk,¡ y = c, y = d ls f?kjs {ks=k dk {ks=kiQy fuEufyf[kr lw=k }kjk izkIr fd;k tkrk gSA

A =

;gk¡ ge {kSfrt ifV~V;ksa ij fopkj djrs gSa tSlk fd vko`Qfr 8-2 esa n'kkZ;k x;k gSA

fVIi.kh ;fn pfpZr oØ dh fLFkfr x-v{k osQ uhps gS] rks tSlk fd vko`Qfr 8-3 esa n'kkZ;k x;k gS] tgk¡ x = a ls x = b rd f (x) < 0 blfy, fn, gq, oØ] x-v{k ,oa dksfV;ksa x = a, x = b ls f?kjs {ks=k dk {ks=kiQy ½.kkRed gks tkrk gS] ijarq ge {ks=kiQy osQ osQoy la[;kRed eku dh gh ppkZ djrs gSaA blfy, ;fn {ks=kiQy ½.kkRed gS rks ge blosQ fujis{k eku] vFkkZr~ dks ysrs gSaA

lkekU;r% ,slk gks ldrk gS fd oØ dk oqQN Hkkx x-v{k osQ Åij gS rFkk oqQN Hkkx x-v{k osQ uhps gS] tSlk fd vko`Qfr 8-4 esa n'kkZ;k x;k gSA ;gk¡ A1 < 0 rFkk A2 > 0 gS] blfy, oØ y = f (x), x-v{k ,oa dksfV;ksa x = a rFkk x = b ls f?kjs {ks=k dk {ks=kiQy A lw=k A = + A2 }kjk izkIr fd;k tkrk gSA

mnkgj.k 1 o`Ùk x2 + y2 = a2 dk {ks=kiQy Kkr dhft,A

gy vko`Qfr 8-5 esa fn, gq, o`Ùk ls f?kjs gq, {ks=k dk oqQy {ks=kiQy

= 4 (fn, gq, oØ] x-v{k ,oa dksfV;ksa x = 0 rFkk x = a ls f?kjs {ks=k AOBA dk {ks=kiQy) [D;ksafd o`Ùk x-v{k ,oa y-v{k nksuksa osQ ifjr% lefer gS]

= (mèokZ/j ifV~V;k¡ ysrs gq,)

= ,

D;ksafd x2 + y2 = a2 ls izkIr gksrk gSA

tSlk fd {ks=k AOBA izFke prqFkk±'k esa lfEefyr gS blfy, y dks /ukRed fy;k tkrk gSA lekdyu djus ij fn, gq, o`Ùk ls f?kjk {ks=kiQy fuEufyf[kr :i esa izkIr gksrk gS%

= =

=

fodYir% tSlk fd vko`Qfr 8-6 esa n'kkZ;k x;k gS {kSfrt ifV~V;ksa dh ppkZ djrs gq, o`Ùk }kjk f?kjs {ks=k dk oqQy {ks=kiQy

= =  

=

=

=

mnkgj.k 2 nh?kZo`Ùk ls f?kjs {ks=k dk {ks=kiQy dk Kkr dhft,A

gy vko`Qfr 8.7 esa nh?kZo`Ùk ls f?kjs {ks=k ABA′B′A dk {ks=kiQy

=

(D;ksafd nh?kZo`Ùk x-v{k ,oa y-v{k nksuksa osQ ifjr% lefer gS)

=

vc = 1 ls izkIr gksrk gS] ijarq {ks=k AOBA izFke prqFkk±'k esa gS blfy, y /ukRed fy;k tkrk gS] blfy, vHkh"V {ks=kiQy

=

(D;ksa?)

gSA

fodYir% tSlk fd vko`Qfr 8-8 esa n'kkZ;k x;k gS {kSfrt ifV~V;ksa dh ppkZ djrs gq, nh?kZo`Ùk dk {ks=kiQy

= = (D;ksa?)

gSA

8.2.1 ,d oØ ,oa ,d js[kk ls f?kjs {ks=k dk {ks=kiQy (The area of the region bounded by a curve and a line)

bl miifjPNsn esa] ge ,d js[kk vkSj ,d o`Ùk] ,d js[kk vkSj ,d ijoy;] rFkk ,d js[kk vkSj ,d nh?kZo`Ùk ls f?kjs {ks=k dk {ks=kiQy Kkr djsaxs mijksDr pfrZr oØksa osQ lehdj.k osQoy izkekf.kd :i esa gh vè;;u fd, tk,¡xs D;ksafd vU; :iksa okys lehdj.k dk mi;ksx bl ikB~;iqLrd osQ vè;;u {ks=k ls ckgj gSaA

mnkgj.k 3 oØ y = x2 ,oa js[kk y = 4 ls f?kjs {ks=k dk {ks=kiQy Kkr dhft,A

gy D;ksafd fn, gq, lehdj.k y = x2 }kjk fu:fir oØ y-v{k osQ ifjr% lefer ,d ijoy; gSA blfy, vko`Qfr 8-9 ls {ks=k AOBA dk vHkh"V {ks=kiQy fuEufyf[kr :i esa izkIr gksrk gS%

= 2 (fn, gq, oØ] y –v{k ,oa js[kkvksa y = 0 rFkk y = 4 ls f?kjs {ks=k BOND dk {ks=kiQy)

= (D;ksa?)

=

;gk¡ geus {kSfrt ifV~V;k¡ yh gSa tSlk fd vko`Qfr 8-9 esa n'kkZ;k x;k gSA

fodYir% {ks=k AOBA dk {ks=kiQy izkIr djus osQ fy, ge PQ tSlh mQèokZ/j ifV~V;k¡ ys ldrs gSa tSlk fd vko`Qfr 8-10 esa n'kkZ;k x;k gSA blosQ fy, ge lehdj.kksa x2 = y ,oa y = 4 dks gy djrs gSa ftlls x = –2 ,oa x = 2 izkIr gksrk gSA

bl izdkj {ks=k AOBA dks oØksa y = x2, y = 4 ,oa dksfV;ksa x = –2 rFkk x = 2 ls f?kjk {ks=k ifjHkkf"kr fd;k tk ldrk gSA

blfy, {ks=k AOBA dk {ks=kiQy

= [y = (¯cnq Q dk y funs±'kkad – ¯cnq P dk y funsZ'kkad) = 4 – x2]

= (D;ksa?)

fVIi.kh mijksDr mnkgj.kksa ls ;g fu"d"kZ fudyrk gS fd fdlh {ks=k dk {ks=kiQy Kkr djus osQ fy, ge mQèokZ/j vFkok {kSfrt ifV~V;ksa esa ls fdlh dks Hkh ys ldrs gSaA blls vkxs ge bu nksuksa ifV~V;ksa esa ls fdlh ,d dh ppkZ djsaxs] mQèokZ/j ifV~V;ksa dks lkekU;r% vf/d izkFkfedrk nh tk,xhA

mnkgj.k 4 izFke prqFkk±'k esa o`Ùk x2 + y2 = 32, js[kk y = x, ,oa x-v{k ls f?kjs {ks=k dk {ks=kiQy Kkr dhft,A

gy fn, gq, lehdj.k gSa%

y = x ... (1)

vkSj x2 + y2 = 32 ... (2)

lehdj.k (1) rFkk (2) dks gy djus ij ge ikrs gSa fd fn;k gqvk o`Ùk ,oa nh gqbZ js[kk ,d nwljs dks izFke prqFkk±'k esa B(4, 4) ij feyrs gSa (vko`Qfr 8-11)A x-v{k osQ Åij BM yEc [khafp,A

blfy,] vHkh"V {ks=kiQy = {ks=k OBMO dk {ks=kiQy + {ks=k BMAB dk {ks=kiQy

vc] {ks=k OBMO dk {ks=kiQy

= = = 8 ... (3)

iqu% {ks=k BMAB dk {ks=kiQy

= =

=

=

= 8 π – (8 + 4π) = 4π – 8 ... (4)

lehdj.k (3) ,oa (4) dk ;ksxiQy Kkr djus ij ge vHkh"V {ks=kiQy A = 4π ikrs gSaA

mnkgj.k 5 nh?kZo`Ùk ,oa dksfV;ksa x = 0 vkSj x = ae, ls f?kjs {ks=k dk {ks=kiQy Kkr dhft,] tgk¡ b2 = a2 (1 – e2) ,oa e < 1 gSA

gy {ks=k BOB′RFSB dk vHkh"V {ks=kiQy fn, gq, nh?kZo`Ùk ,oa js[kkvksa x = 0 rFkk x = ae ls f?kjk gqvk gS(vko`Qfr 8-12)A

è;ku nhft, fd {ks=k BOB′RFSB dk {ks=kiQy

= =

=

=

=

iz'ukoyh 8-1

1. oØ y2 = x, js[kkvksa x = 1, x = 4 ,oa x-v{k ls f?kjs {ks=k dk izFke ikn esa {ks=kiQy Kkr dhft,A

2. izFke prqFkk±'k esa oØ y2 = 9x, x = 2, x = 4 ,oa x-v{k ls f?kjs {ks=k dk {ks=kiQy Kkr dhft,A

3. izFke prqFkk±'k esa x2 = 4y, y = 2, y = 4 ,oa y-v{k ls f?kjs {ks=k dk {ks=kiQy Kkr dhft,A

4. nh?kZo`Ùk ls f?kjs {ks=k dk {ks=kiQy Kkr dhft,A

5. nh?kZo`Ùk ls f?kjs {ks=k dk {ks=kiQy Kkr dhft,A

6. izFke prqFkk±'k esa o`Ùk x2 + y2 = 4, js[kk x =y ,oa x-v{k }kjk f?kjs {ks=k dk {ks=kiQy Kkr dhft,A

7. Nsnd js[kk }kjk o`Ùk x2 + y2 = a2 osQ NksVs Hkkx dk {ks=kiQy Kkr dhft,A

8. ;fn oØ x = y2 ,oa js[kk x = 4 ls f?kjk gqvk {ks=kiQy js[kk x = a }kjk nks cjkcj Hkkxksa esa foHkkftr gksrk gS rks a dk eku Kkr dhft,A

9. ijoy; y = x2 ,oa y = ls f?kjs {ks=k dk {ks=kiQy Kkr dhft,A

10. oØ x2 = 4y ,oa js[kk x = 4y – 2 ls f?kjs {ks=k dk {ks=kiQy Kkr dhft,A

11. oØ y2 = 4x ,oa js[kk x = 3 ls f?kjs {ks=k dk {ks=kiQy Kkr dhft,A

iz'u 12 ,oa 13 esa lgh mÙkj dk p;u dhft,%

12. izFke prqFkk±'k esa o`Ùk x2 + y2 = 4 ,oa js[kkvksa x = 0, x = 2 ls f?kjs {ks=k dk {ks=kiQy gS%

(A) π (B) (C) (D)

13. oØ y2 = 4x, y-v{k ,oa js[kk y = 3 ls f?kjs {ks=k dk {ks=kiQy gS%

(A) 2 (B) (C) (D)

8.3 nks oØksa osQ eè;orhZ {ks=k dk {ks=kiQy (Area Between Two Curves)

yScfu”k dh psruk ,oa varKkZu dh lPpkbZ osQ iQyLo:i fdlh {ks=k dks izkjafHkd {ks=kiQy dh c`gr~ la[;k esa ifV~V;k¡ dkVdj vkSj bu izkjafHkd {ks=kiQyksa dk ;ksxiQy Kkr dj] {ks=kiQy osQ ifjdyu dh fØ;k lekdyu dgykrh gSA dYiuk dhft,] gesa nks oØ y = f (x) vkSj y = g (x) fn, gq, gSa tgk¡ [a, b]esa f(x) ≥ g(x) tSlk fd vko`Qfr 8-13 esa n'kkZ;k x;k gSA fn, gq, oØksa osQ lehdj.k ls y dk mHk;fu"B eku ysrs gq, bu nksuksa oØksa osQ izfrPNsnd ¯cnq x = a rFkk x = b }kjk ns; gSaA

lekdyu osQ lw=k dk LFkkiu djus osQ fy, izkjafHkd {ks=kiQy dks mQèokZ/j ifV~V;ksa osQ :i esa ysuk lqfo/ktud gSA tSlk fd vko`Qfr 8-13 esa n'kkZ;k x;k gSA izkjafHkd iV~Vh dh m¡QpkbZ
f(x) – g(x) ,oa pkSM+kbZ dx gS] blfy, izkjafHkd {ks=kiQy

dA = [f(x) – g(x)] dx, rFkk oqQy {ks=kiQy A =

fodYir%

A = [oØ y = f (x), x-v{k rFkk js[kkvksa x = a, x = b ls f?kjs {ks=k dk {ks=kiQy]

– [oØ y = g (x), x-v{k ,oa js[kkvksa x = a, x = b ls f?kjs {ks=k dk {ks=kiQy]

= =tgk¡ [a, b] esa f (x) ≥ g (x)

;fn [a, c] esa f (x) ≥ g (x) rFkk [c, b] esa f (x) ≤ g (x) tgk¡ a < c < b tSlk fd vko`Qfr 8-14 esa n'kkZ;k x;k gS] rks oØksa ls f?kjs {ks=kksa dk {ks=kiQy fuEufyf[kr izdkj fy[kk tk ldrk gS%

{ks=kiQy = {ks=k ACBDA dk {ks=kiQy + {ks=k BPRQB dk {ks=kiQy

=

mnkgj.k 6 nks ijoy;ksa y = x2 ,oa y2 = x ls f?kjs {ks=k dk {ks=kiQy Kkr dhft,A

gy tSlk fd vko`Qfr 8-15 esa n'kkZ;k x;k gS] bu nksuksa ijoy;ksa osQ izfrPNsnd ¯cnq O (0, 0) ,oa A (1, 1) gSA

blfy, Nk;kafoQr {ks=k dk vHkh"V {ks=kiQy

=

= =

mnkgj.k 7 x-v{k osQ Åij rFkk o`Ùk x2 + y2 = 8x ,oa ijoy; y2 = 4x osQ eè;orhZ {ks=k dk {ks=kiQy Kkr dhft,A

gy o`Ùk dk fn;k gqvk lehdj.k x2 + y2 = 8x, (x – 4)2 + y2 = 16 osQ :i esa vfHkO;Dr fd;k tk ldrk gSA bl o`Ùk dk osaQnz ¯cnq (4, 0) gS rFkk f=kT;k 4 bdkbZ gSA ijoy; y2 = 4x osQ lkFk blosQ izfrPNsn ls izkIr gksrk gS%

x2 + 4x = 8x

vFkok x2 – 4x = 0

vFkok x (x – 4) = 0

vFkok x = 0, x = 4

bl izdkj bu nks oØksa osQ izfrPNsn ¯cnq O(0, 0) ,oa x-v{k ls Åij P(4,4) gSaA

vko`Qfr 8-16 ls x-v{k ls mij bu nksuksa oØksa osQ eè; lfEefyr {ks=k OPQCO dk {ks=kiQy

= ({ks=k OCPO dk {ks=kiQy) + ({ks=k PCQP dk {ks=kiQy)

=

= (D;ksa\)

=

=

=  =

mnkgj.k 8 vko`Qfr 8-17 esa AOBA izFke prqFkk±'k esa nh?kZo`Ùk 9x2 + y2 = 36 dk ,d Hkkx gS ftlesa OA = 2 bdkbZ rFkk OB = 6 bdkbZ gSA y?kq pki AB ,oa thok AB osQ eè;orhZ {ks=k dk {ks=kiQy Kkr dhft,A

gy nh?kZo`Ùk dk fn;k gqvk lehdj.k 9x2 + y2 = 36, vFkkZr~ vFkok osQ :i esa vfHkO;Dr fd;k tk ldrk gS vkSj blfy, bldk vkdkj vko`Qfr 8-17 esa fn, gq, vkdkj tSlk gSA

blosQ vuqlkj] thok AB dk lehdj.k gS%

y – 0 =

vFkok y = – 3 (x – 2)

vFkok y = – 3x + 6

vko`Qfr 8-17 esa n'kkZ;s Nk;kafoQr {ks=k dk {ks=kiQy

= (D;ksa\)

=

= = 3π – 6

mnkgj.k 9 lekdyu dk mi;ksx djrs gq, ,d ,sls f=kHkqt dk {ks=kiQy Kkr dhft, ftlosQ 'kh"kZ (1, 0), (2, 2) ,oa (3, 1) gSaA

gy eku yhft, A(1, 0), B(2, 2) ,oa C(3, 1) f=kHkqt ABC osQ 'kh"kZ gSa (vko`Qfr 8-18)

∆ ABC dk {ks=kiQy = ∆ ABD dk {ks=kiQy + leyac prqHkqZt BDEC dk {ks=kiQy – ∆ AEC dk {ks=kiQy

vc Hkqtk,¡ AB, BC ,oa CA osQ lehdj.k Øe'k%

y = 2(x – 1), y = 4 – x, y = (x – 1) gSaA

vr% ∆ ABC dk {ks=kiQy

=

=

=

– =

mnkgj.k 10 nks o`Ùkksa x2 + y2 = 4 ,oa (x – 2)2 + y2 = 4 osQ eè;orhZ {ks=k dk {ks=kiQy Kkr dhft,A

gy fn, gq, o`Ùkksa osQ lehdj.k gSa%

x2 + y2 = 4 ... (1)

vkSj (x – 2)2 + y2 = 4 ... (2)

lehdj.k (1) ,slk o`Ùk gS ftldk osaQnz ewy ¯cnq O ij gS vksj ftldh f=kT;k 2 bdkbZ gSA lehdj.k (2) ,d ,slk o`Ùk gS ftldk osaQnz C(2, O) gS vkSj ftldh f=kT;k 2 bdkbZ gSA lehdj.k (1) vkSj (2) dks gy djus ij ge ikrs gSa%

(x –2)2 + y2 = x2 + y2

vFkok x2 – 4x + 4 + y2 = x2 + y2

vFkok x = 1 ftlls y = izkIr gksrk gSA

vr% fn, gq, o`Ùkksa osQ izfrPNsnu ¯cnq A(1, ) vkSj A′(1, ) gS] tSlk vko`Qfr 8-19 esa n'kkZ;k x;k gSA

o`Ùkksa osQ eè;orhZ {ks=k OACA′O dk vHkh"V

{ks=kiQy = 2 [{ks=k ODCAO dk {ks=kiQy] (D;ksa?)

= 2 [{ks=k ODAO dk {ks=kiQy + {ks=k DCAD dk {ks=kiQy]

=

= (D;ksa?)

= +

=

=

=

=

=

iz'ukoyh 8-2

1. ijoy; x2 = 4y vkSj o`Ùk 4x2 + 4y2 = 9 osQ eè;orhZ {ks=k dk {ks=kiQy Kkr dhft,A

2. oØksa (x – 1)2 + y2 = 1 ,oa x2 + y2 = 1 ls f?kjs {ks=k dk {ks=kiQy Kkr dhft,A

3. oØksa y = x2 + 2, y = x, x = 0 ,oa x = 3 ls f?kjs {ks=k dk {ks=kiQy Kkr dhft,A

4. lekdyu dk mi;ksx djrs gq, ,d ,sls f=kHkqt dk {ks=kiQy Kkr dhft, ftlosQ 'kh"kZ
(– 1, 0), (1, 3) ,oa (3, 2) gSaA

5. lekdyu dk mi;ksx djrs gq, ,d ,sls f=kdks.kh; {ks=k dk {ks=kiQy Kkr dhft, ftldh Hkqtkvksa osQ lehdj.k y = 2x + 1, y = 3x + 1 ,oa x = 4 gSaA

iz'u 6 ,oa 7 esa lgh mÙkj dk p;u dhft,%

6. o`Ùk x2 + y2 = 4 ,oa js[kk x + y = 2 ls f?kjs NksVs Hkkx dk {ks=kiQy gS%

(A) 2 (π – 2) (B) π – 2 (C) 2π – 1 (D) 2 (π + 2)

7. oØksa y2 = 4x ,oa y = 2x osQ eè;orhZ {ks=k dk {ks=kiQy gS%

(A) (B) (C) (D)

fofo/ mnkgj.k

mnkgj.k 11 ijoy; y2 = 4ax vkSj mlosQ ukfHkyac ls f?kjs {ks=k dk {ks=kiQy Kkr dhft,A

gy vko`Qfr 8.20 ls] ijoy; y2 = 4ax dk 'kh"kZ ewy ¯cnq ij gSA ukfHkyac thok LSL′ dk lehdj.k x = a gSA fn;k gqvk ijoy; x-v{k osQ ifjr% lefer gSA

{ks=k OLL′O dk vHkh"V {ks=kiQy = 2 (+{ks=k OLSO dk {ks=kiQy)

= =

=

= = =

mnkgj.k 12 js[kk y = 3x + 2, x-v{k ,oa dksfV;ksa x = –1 ,oa x = 1 ls f?kjs {ks=k dk {ks=kiQy Kkr dhft,A

gy tSlk fd vko`Qfr 8-21 esa n'kkZ;k x;k gS] js[kk
y = 3x + 2, x-v{k dks x = ij feyrh gS vkSj osQ fy, bldk vkys[k x-v{k osQ uhps gS rFkk osQ fy, bldk vkys[k x-v{k ls Åij gSA

vHkh"V {ks=kiQy = {ks=k ACBA dk {ks=kiQy + {ks=k ADEA dk {ks=kiQy

=

= =

mnkgj.k 13 x = 0 ,oa x = 2π osQ eè; oØ
y = cosx ls f?kjs {ks=k dk {ks=kiQy Kkr dhft,A

gy vko`Qfr 8-22 ls] vHkh"V {ks=kiQy

= {ks=k OABO dk {ks=kiQy + {ks=k BCDB dk {ks=kiQy + {ks=k DEFD dk {ks=kiQy

blfy, vHkh"V {ks=kiQy

=

= = 1 + 2 + 1 = 4

mnkgj.k 14 fl¼ dhft, fd oØ y2 = 4x ,oa x2 = 4y, js[kkvksa x = 0, x = 4, y = 4 ,oa y = 0 ls f?kjs oxZ osQ {ks=kiQy dks rhu cjkcj Hkkxksa esa foHkkftr djrs gSaA

gy è;ku nhft, fd ijoy;ksa y2 = 4x ,oa x2 = 4y osQ izfrPNsn ¯cnq (0,0) ,oa (4,4) gSa tSlk fd vko`Qfr 8-23 esa n'kkZ;k x;k gSA

vc oØksa y2 = 4x ,oa x2 = 4y ls f?kjs {ks=k OAQBO dk {ks=kiQy

= =

= ... (1)

iqu% oØksa x2 = 4y, x = 0, x = 4 ,oa x-v{k ls f?kjs {ks=k OPQAO dk {ks=kiQy =... (2)

blh izdkj oØ y2 = 4x, y-v{k, y = 0 ,oa y = 4 ls f?kjs {ks=k OBQRO dk {ks=kiQy=  ... (3)

lehdj.kksa (1)] (2) rFkk (3) ls ;g fu"d"kZ fudyrk gS fd

{ks=k OAQBO dk {ks=kiQy = {ks=k OPQAO dk {ks=kiQy = {ks=k OBQRO dk {ks=kiQy vFkkZr~] ijoy;ksa y2 = 4x ,oa x2 = 4y ls f?kjk {ks=kiQy fn, gq, oxZ osQ {ks=kiQy dks rhu cjkcj Hkkxksa esa foHkkftr djrk gSA

mnkgj.k 15 {ks=k {(x, y) : 0 ≤ y ≤ x2 + 1, 0 ≤ y ≤ x + 1, 0 ≤ x ≤ 2}dk {ks=kiQy Kkr dhft,A

gy vkb, loZizFke ge ml {ks=k dk js[kkfp=k rS;kj djsa ftldk gesa {ks=kiQy Kkr djuk gSA ;g {ks=k fuEufyf[kr {ks=kksa dk eè;orhZ {ks=k gS%

A1 = {(x ,y) : 0 ≤ y ≤ x2+ 1}

A2 = {(x, y) : 0 ≤ y ≤ x + 1}

vkSj A3 = {(x, y) : 0 ≤ x ≤ 2}

oØksa y = x2 + 1 ,oa y = x + 1 osQ izfrPNsn ¯cnq P(0, 1) ,oa Q(1, 2) gSaA vko`Qfr 8-24 ls] vHkh"V {ks=k] Nk;kafdr {ks=k OPQRSTO gS ftldk {ks=kiQy

= {ks=k OTQPO dk {ks=kiQy + {ks=k TSRQT dk {ks=kiQy

= (D;ksa?)

=

= =

vè;k; 8 ij fofo/ iz'ukoyh

1. fn, gq, oØksa ,oa js[kkvksa ls f?kjs {ks=k dk {ks=kiQy Kkr dhft,%

(i) y = x2; x = 1, x = 2 ,oa x-v{k

(ii) y = x4; x = 1, x = 5 ,o x-v{k

2. oØksa y = x ,oa y = x2 osQ eè;orhZ {ks=k dk {ks=kiQy Kkr dhft,A

3. izFke prqFkk±'k esa lfEefyr ,oa y = 4x2, x = 0, y = 1 rFkk y = 4 ls f?kjs {ks=k dk {ks=kiQy Kkr dhft,A

4. y = dk xzki+Q [khafp, ,oa dk eku Kkr dhft,A

5. x = 0 ,oa x = 2π rFkk oØ y = sinx ls f?kjs {ks=k dk {ks=kiQy Kkr dhft,A

6. ijoy; y2 = 4ax ,oa js[kk y = mx ls f?kjs {ks=k dk {ks=kiQy Kkr dhft,A

7. ijoy; 4y = 3x2 ,oa js[kk 2y = 3x + 12 ls f?kjs {ks=k dk {ks=kiQy Kkr dhft,A

8. nh?kZo`Ùk ,oa js[kk ls f?kjs y?kq {ks=k dk {ks=kiQy Kkr dhft,A

9. nh?kZo`Ùk ,oa js[kk ls f?kjs y?kq {ks=k dk {ks=kiQy Kkr dhft,A

10. ijoy; x2 = y, js[kk y = x + 2 ,oa x-v{k ls f?kjs {ks=k dk {ks=kiQy Kkr dhft,A

11. lekdyu fof/ dk mi;ksx djrs gq, oØ ls f?kjs {ks=k dk {ks=kiQy Kkr dhft,A

[laosQr : vko';d {ks=k] js[kkvksa x + y = 1, x– y = 1, – x + y = 1 ,oa – x – y = 1 ls f?kjk gS]

12. oØksa {(x, y) : y ≥ x2 rFkk y = } ls f?kjs {ks=k dk {ks=kiQy Kkr dhft,A

13. lekdyu fof/ dk mi;ksx djrs gq, ,d ,sls f=kHkqt ABC, dk {ks=kiQy Kkr dhft, ftlosQ 'kh"kks± osQ funsZ'kkad A(2, 0), B (4, 5) ,oa C (6, 3) gSaA

14. lekdyu fof/ dk mi;ksx djrs gq,] js[kkvksa 2x + y = 4, 3x – 2y = 6 ,oa x – 3y + 5 = 0 ls f?kjs {ks=k dk {ks=kiQy Kkr dhft,A

15. {ks=k {(x, y) : y2 ≤ 4x, 4x2 + 4y2 ≤ 9}dk {ks=kiQy Kkr dhft,A

16 ls 20 rd iz'uksa esa lgh mÙkj dk p;u dhft,%

16. oØ y = x3, x-v{k ,oa dksfV;ksa x = – 2, x = 1 ls f?kjs {ks=k dk {ks=kiQy gS%

(A) – 9 (B) (C) (D)

17. oØ y = x |x|, x-v{k ,oa dksfV;ksa x = – 1 rFkk x = 1 ls f?kjs {ks=k dk {ks=kiQy gS%

(A) 0 (B) (C) (D)

[laosQr : y = x2 ;fn x > 0 ,oa y = – x2 ;fn x < 0]

18. {ks=k y2 ≥ 6x vkSj o`Ùk x2 + y2 = 16 esa lfEefyr {ks=k dk {ks=kiQy gS%

(A) (B) (C) (D)

19. y-v{k, y = cos x ,oa y = sin x, ls f?kjs {ks=k dk {ks=kiQy gS%

(A) (B) (C) (D)