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

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)

mnkgj.k 3 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 4 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

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)