KEY CONCEPTS. satisfies the differential equation da. = 0. Note : If F (x) is any integral of f (x) then, x a


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1 KEY CONCEPTS THINGS TO REMEMBER :. The re ounded y the curve y = f(), the is nd the ordintes t = & = is given y, A = f () d = y d.. If the re is elow the is then A is negtive. The convention is to consider the mgnitude only i.e. A = y d in this cse.. Are etween the curves y = f () & y = g () etween the ordintes t = & = is given y, A = f () d g () d = [ f () g () ] d.. Averge vlue of function y = f () w.r.t. over n intervl is defined s : y (v) = f () d. 5. The re functiona stisfies the differentil eqution da = f () with initil conditiona d =. Note : If F () is ny integrl of f () then, A = f () d = F () + c A = = F () + c c = F () hence A = F () F (). Finlly y tking = we get, A = F () F (). 6. CURVE TRACING : The following outline procedure is to e pplied in Sketching the grph of function y = f () which in turn will e etremely useful to quickly nd correctly evlute the re under the curves. () Symmetry : The symmetry of the curve is judged s follows : (i) If ll the powers of y in the eqution re even then the curve is symmetricl out the is of. (ii) If ll the powers of re even, the curve is symmetricl out the is of y. (iii) If powers of & y oth re even, the curve is symmetricl out the is of s well s y. (iv) If the eqution of the curve remins unchnged on interchnging nd y, then the curve is symmetricl out y =. (v) If on interchnging the signs of & y oth the eqution of the curve is unltered then there is symmetry in opposite qudrnts. () Find dy/d & equte it to zero to find the points on the curve where you hve horizontl tngents. (c) Find the points where the curve crosses the is & lso the yis. (d) Emine if possile the intervls when f () is incresing or decresing. Emine wht hppens to y when or. 7. USEFUL RESULTS : (i) Whole re of the ellipse, / + y / = is π. (ii) Are enclosed etween the prols y = & = y is 6/. (iii) Are included etween the prol y = & the line y = m is 8 / m. EXERCISE I Q. Find the re ounded on the right y the line + y =, on the left y the prol y = nd elow y the is. Q. Find the re of the region ounded y the curves, y = ² + ; y = ; = & =. Pge 9 of Are Under Curve
2 Q. Find the re of the region {(, y) : y ² +, y +, }. Q. Find the vlue of c for which the re of the figure ounded y the curves y = sin, the stright lines = π/6, = c & the sciss is is equl to /. Q.5 The tngent to the prol y = hs een drwn so tht the sciss of the point of tngency elongs to the intervl [, ]. Find for which the tringle ounded y the tngent, the is of ordintes & the stright line y = hs the gretest re. Q.6 Compute the re of the region ounded y the curves y = e.. ln & y = ln /(e. ) where ln e=. Q.7 A figure is ounded y the curves y = sin π, y =, = & =. At wht ngles to the positive is stright lines must e drwn through (, ) so tht these lines prtition the figure into three prts of the sme size. Q.8 Find the re of the region ounded y the curves, y = log e, y = sin π & =. Q.9 Find the re ounded y the curves y = nd y =. Also find the rtio in which the yis divided this re. Q. If the re enclosed y the prols y = nd y = is 8 sq. units. Find the vlue of ''. Q. The line + y = divides the re enclosed y the curve, 9 + y 8 6y = into two prts. Find the rtio of the lrger re to the smller re. Q. Find the re of the region enclosed etween the two circles ² + y² = & ( )² + y² = Q. Find the vlues of m (m > ) for which the re ounded y the line y = m + nd = y y is, (i) 9/ squre units & (ii) minimum. Also find the minimum re. Q. Find the rtio in which the re enclosed y the curve y = cos ( π/) in the first qudrnt is divided y the curve y = sin. Q.5 Find the re enclosed etween the curves : y = log e ( + e), = log e (/y) & the is. Q.6 Find the re of the figure enclosed y the curve (y rc sin ) =. Q.7 For wht vlue of '' is the re ounded y the curve y = + + nd the stright line y =, = & = the lest? Q.8 Find the positive vlue of '' for which the prol y = + isects the re of the rectngle with vertices (, ), (, ), (, + ) nd (, + ). Q.9 Compute the re of the curviliner tringle ounded y the yis & the curve, y = tn & y = (/) cos. Q. Consider the curve C : y = sin sin, C cuts the is t (, ), ( π, π). A : The re ounded y the curve C & the positive is etween the origin & the ordinte t =. A : The re ounded y the curve C & the negtive is etween the ordinte = & the origin. Prove tht A + A + 8 A A =. Q. Find the re ounded y the curve y = e ; y = nd = c where c is the coordinte of the curve's inflection point. Q. Find the vlue of 'c' for which the re of the figure ounded y the curve, y = 8 5, the stright lines = & = c & the sciss is is equl to 6/. Q. Find the re ounded y the curve y² = & = y. Q. Find the re ounded y the curve y = e, the is, nd the line = c where y (c) is mimum. Q.5 Find the re of the region ounded y the is & the curves defined y, y = tn, π / π / y = cot, π / 6 π / Pge of Are Under Curve
3 EXERCISE II Q. In wht rtio does the is divide the re of the region ounded y the prols y = ² & y = ²? Q. Find the re ounded y the curves y = & y =. Q. Sketch the region ounded y the curves y = 5 & y = & find its re. Q. Find the eqution of the line pssing through the origin nd dividing the curviliner tringle with verte t the origin, ounded y the curves y =, y = nd = into two prts of equl re. Q.5 Consider the curve y = n where n > in the st qudrnt. If the re ounded y the curve, the is nd the tngent line to the grph of y = n t the point (, ) is mimum then find the vlue of n. Q.6 Consider the collection of ll curve of the form y = tht pss through the the point (, ), where nd re positive constnts. Determine the vlue of nd tht will minimise the re of the region ounded y y = nd is. Also find the minimum re. Q.7 In the djcent grphs of two functions y = f() nd y = sin re given. y = sin intersects, y = f() t A (, f()); B(π, ) nd C(π, ). A i (i =,,,) is the re ounded y the curves y = f () nd y = sin etween = nd = ; i =, etween = nd = π; i =, etween = π nd = π; i =. If A = sin + ( )cos, determine the function f(). Hence determine nd A. Also clculte A nd A. Q.8 Consider the two curves y = /² & y = /[ ( )]. (i) At wht vlue of ( > ) is the reciprocl of the re of the fig. ounded y the curves, the lines = & = equl to itself? (ii) At wht vlue of ( < < ) the re of the figure ounded y these curves, the lines = & = equl to /. Q.9 ln c Show tht the re ounded y the curve y =, the is nd the verticl line through the mimum point of the curve is independent of the constnt c. Q. For wht vlue of '' is the re of the figure ounded y the lines, y =, y = 5? Q. Compute the re of the loop of the curve y² = ² [( + )/( )]. Q. Find the vlue of K for which the re ounded y the prol y = + nd the line y = K + is lest. Also find the lest re. Q. Let A n e the re ounded y the curve y = (tn ) n & the lines =, y = & = π/. Prove tht for n >, A n + A n = /(n ) & deduce tht /(n + ) < A n < /(n ). Q. If f () is monotonic in (, ) then prove tht the re ounded y the ordintes t = ; = ; y = f () + nd y = f (c), c (, ) is minimum when c =. Hence if the re ounded y the grph of f () = +, the stright lines =, = nd the is is minimum then find the vlue of ''. Q.5 Consider the two curves C : y = + cos & C : y = + cos ( α) for α, π ; [, π]. Find the vlue of α, for which the re of the figure ounded y the curves C, C & = is sme s tht of the figure ounded y C, y = & = π. For this vlue of α, find the rtio in which the line y = divides the re of the figure y the curves C, C & = π. Q.6 Find the re ounded y y² = ( + ), y² = ( ) & y = ove is of. Q.7 Compute the re of the figure which lies in the first qudrnt inside the curve Pge of Are Under Curve
4 ² + y² = ² & is ounded y the prol ² = y & y² = ( > ). Q.8 Consider squre with vertices t (, ), (, ), (, ) & (, ). Let S e the region consisting of ll points inside the squre which re nerer to the origin thn to ny edge. Sketch the region S & find its re. Q.9 Find the whole re included etween the curve ² y² = ² (y² ²) & its symptotes (symptotes re the lines which meet the curve t infinity). Q. For wht vlues of [, ] does the re of the figure ounded y the grph of the function y = f () nd the stright lines =, = & y = f() is t minimum & for wht vlues it is t mimum if f () =. Find lso the mimum & the minimum res. Q. Find the re enclosed etween the smller rc of the circle ² + y² + y = & the prol y = ² + +. Q. D r w n e t n d c l e n g r p h o f t h e f u n c t i o n f D r w n e t n d c l e n g r p h o f t h e f u n c t i o n f () = cos ( ), [, ] nd find the re enclosed etween the grph of the function nd the is s vries from to. Q. Let C & C e two curves pssing through the origin s shown in the figure. A curve C is sid to "isect the re" the region etween C & C, if for ech point P of C, the two shded regions A & B shown in the figure hve equl res. Determine the upper curve C, given tht the isecting curve C hs the eqution y = & tht the lower curve C hs the eqution y = /. Q. For wht vlues of [, ] does the re of the figure ounded y the grph of the function y = f () & the stright lines =, =, y = f() hve the gretest vlue nd for wht vlues does it hve the lest vlue, if, f() = α + β, α, β R with α >, β >. Q.5 Given f () = t e (logsec t sec y = f () nd y = g () etween the ordintes = nd = π. t)dt ; g () = e tn. Find the re ounded y the curves EXERCISE III Q. Let f () = Mimum {, ( ), ( )}, where. Determine the re of the region ounded y the curves y = f (), is, = & =. [ JEE '97, 5 ] Q. Indicte the region ounded y the curves = y, y = + nd is nd otin the re enclosed y them. [ REE '97, 6 ] Q. Let C & C e the grphs of the functions y = & y =, respectively. Let C e the grph of function y = f (),, f() =. For point P on C, let the lines through P, prllel to the es, meet C & C t Q & R respectively (see figure). If for every position of P (on C ), the res of the shded regions OPQ & ORP re equl, determine the function f(). [JEE '98, 8] Q. Indicte the region ounded y the curves y = ln & y = nd otin the re enclosed y them. [ REE '98, 6 ] Q.5 () For which of the following vlues of m, is the re of the region ounded y the curve y = nd the line y = m equls 9/? (A) (B) (C) (D) for () Let f() e continuous function given y f() = + + for > Find the re of the region in the third qudrnt ounded y the curves, = y nd Pge of Are Under Curve
5 y = f() lying on the left of the line 8 + =. [ JEE '99, + (out of ) ] Q.6 Find the re of the region lying inside + (y ) = nd outside c + y = c where c =. [REE '99, 6] Q.7 Find the re enclosed y the prol (y ) =, the tngent to the prol t (, ) nd the is. [REE,] Q.8 Let nd for j =,,,...n, let S j e the re of the region ounded y the y is nd the curve jπ ( j+ ) π e y = siny, y. Show tht S, S, S,...S n re in geometric progression. Also, find their sum for = nd = π. [JEE', 5] Q.9 The re ounded y the curves y = nd y = + is (A) (B) (C) (D) [JEE', (Scr)] Q. Find the re of the region ounded y the curves y =, y = nd y =, which lies to the right of the line =. [JEE ', (Mins)] Q. If the re ounded y y = nd = y, >, is, then = (A) (B) (C) (D) [JEE ', (Scr)] Q.() The re ounded y the prols y = ( + ) nd y = ( ) nd the line y = / is (A) sq. units (B) /6 sq. units (C) / sq. units (D) / sq. units [JEE '5 (Screening)] () Find the re ounded y the curves = y, = y nd y =. f ( ) + (c) If f () = +, f () is qudrtic function nd its mimum vlue occurs t c c f () c + c point V. A is point of intersection of y = f () with is nd point B is such tht chord AB sutends right ngle t V. Find the re enclosed y f () nd chord AB. [JEE '5 (Mins), + 6] Q. Mtch the following π cos sin (i) (sin ) (cos cot log(sin ) )d (A) (ii) Are ounded y y = nd = 5y (B) (iii) Cosine of the ngle of intersection of curves y = log nd y = is (C) 6 ln (D) / [JEE 6, 6] ANSWER EXERCISE I Q. 5/6 sq. units Q. / sq. units Q. /6 sq. units Q. c = π 6 or π Q 7. π tn π Q 9. π ; π π + ; π tn π Q 5. =, A( ) = 8 Q 8. 8 Q 6. (e 5)/ e sq. units sq. units Q. = 9 Q. π + π Pge of Are Under Curve
6 Q. π sq. units Q. (i) m =, (ii) m = ; Amin = / Q. Q 5. sq. units Q 6. π/ Q 7. = / Q 8. Q 9. + l n sq. units 8 7 / Q. e Q. C = or ( ) Q. / Q. ( e / ) Q 5. ln EXERCISE II Q. : Q. 8/5 sq. units Q. (5 π )/ sq. units Q. y = / Q 5. + Q 6. = /8, A minimum = sq. units Q 7. f() = sin, = ; A = sin; A = π sin; A = (π ) sq. units Q 8. = + e, = + e Q.9 / Q. = 8 or ( 6 ) 5 Q. (π/) sq. units Q. K =, A = / Q. = 8 8 / Q 5. α = π/, rtio = : Q 6. ( ) ( ) Q sin rc sq. units Q 8. ( 6 ) Q 9. Q. = / gives minim, A = π π ; = gives locl mim A() = ; = gives mimum vlue, A() = π/ Q. 8 + π Q. ( ) sq. units Q. (6/9) Q. for =, re is gretest, for = /, re is lest Q5. e π log sq. units EXERCISE III Q. 7/7 Q. 5/6 sq. units Q. f() = Q. 7/ π Q.5 () B, D () 57/9 ; = ; = Q.6 π sq. units π Sj Q.7 9 sq. units Q.8 = e ; S S j+ e = π + + for =, = π, S = π ( e + ) nd r = π π + Q.9 B Q. sq. units Q. B 5 Q. () D ; () sq. units ; (c) sq. units Q. (i) A, (ii) D, (iii) A Pge of Are Under Curve
7 EXERCISE IV. The re ounded y the curve = y, is nd the line = is (A) (B) (C). The re ounded y the is nd the curve y = is (A) (B) (C) (D) (D) 8 Pge 5 of Are Under Curve. The re ounded y the curve y = sin with is in one rc of the curve is (A) (B) (C) (D). The re contined etween the curve y =, the verticl line =, = ( > ) nd is is (A) log (B) log (C) log (D) log 5. The re of the closed figure ounded y the curves y =, y = & y = is: (A) 9 (B) 8 9 (C) 6 9 (D) none 6. The re of the closed figure ounded y the curves y = cos ; y = + π & = π is (A) π + (B) π (C) π + 7. The re included etween the curve y = ( ) & its symptote is: (A) π 8. The re ounded y ² + y² = & y = sin π (A) π π (B) π π (D) π (B) π (C) π (D) none (C) π in the upper hlf of the circle is: 8 π (D) none 9. The re of the region enclosed etween the curves 7 + 9y + 9 = nd y + 7 = is: (A) (B) (C) 8 (D) 6. The re ounded y the curves y = ( ln ); = e nd positive Xis etween = e nd = e is : e (A) 5 e e (B) 5 e e e (C) 5 5e (D). The re enclosed etween the curves y = log e ( + e), = log e y nd the is is (A) (B) (C) (D) none of these e
8 . The re ounded y the curves + y = nd + y = is (A) (B) 6 (C) (D) none of these. The re ounded y is, curve y = f(), nd lines =, = is equl to ( + ) for ll >, then f() is (A) ( ) (B) ( + ) (C) ( + ) (D) / (+ ). The re of the region for which < y < nd > is (A) ( ) d (B) ( ) d (C) ( ) d (D) ( 5. The re ounded y y =, y = [ + ], nd the yis is (A) / (B) / (C) (D) 7/ 6. The re ounded y the curve = cos t, y = sin t is (A) π 8 (B) π 6 (C) π (D) π 7. If A is the re enclosed y the curve y =, is nd the ordintes =, = ; nd A is the re enclosed y the curve y =, is nd the ordintes =, =, then (A) A = A (B) A = A (C) A = A (D) A = A 8. The re ounded y the curv e y = f(), is nd the ordintes = nd = is ( ) sin ( + ), R, then f() = (A) ( ) cos ( + ) (B) sin ( + ) (C) sin ( + ) + ( ) cos ( + ) (D) none of these 9. Find the re of the region ounded y the curves y = +, y =, = nd =. (A) sq. unit (B) sq. unit (C) sq. unit (D) none of these. The res of the figure into which curve y = 6 divides the circle + y = 6 re in the rtio (A) (B) π 8π + (C) π + 8π (D) none of these. The tringle formed y the tngent to the curve f() = + t the point (, ) nd the coordinte es, lies in the first qudrnt. If its re is, then the vlue of is [IIT  ] (A) (B) (C) (D) EXERCISE V. Find the re of the region ounded y the curve y = y nd the yis.. Find the vlue of c for which the re of the figure ounded y the curves y = sin, the stright lines = π/6, = c & the sciss is is equl to /.. For wht vlue of '' is the re ounded y the curve y = + + nd the stright line y =, = & = the lest?. Find the re of the region ounded in the first qudrnt y the curve C: y = tn, tngent drwn to ) d Pge 6 of Are Under Curve
9 C t = π nd the is. 5. Find the vlues of m (m > ) for which the re ounded y the line y = m + nd = y y is, (i) 9/ squre units & (ii) minimum. Also find the minimum re. 6. Consider the two curves y = /² & y = /[ ( )]. (i) At wht vlue of ( > ) is the reciprocl of the re of the figure ounded y the curves, the lines = & = equl to itself? (ii) At wht vlue of ( < < ) the re of the figure ounded y these curves, the lines = & = equl to /. 7. A norml to the curve, + α y + = t the point whose sciss is, is prllel to the line y =. Find the re in the first qudrnt ounded y the curve, this norml nd the is of ' '. 8. Find the re etween the curve y ( ) = & its symptotes. 9. Drw net & clen grph of the function f () = cos ( ), [, ] & find the re enclosed etween the grph of the function & the is s vries from to.. Find the re of the loop of the curve, y = ( ).. Let nd for j =,,,..., n, let S j e the re of the region ounded y the yis nd the curve e y = sin y, j π ( j +) π y. Show tht S, S, S,..., S n re in geometric progression. Also, find their sum for = nd = π. [IIT , 5]. Find the re of the region ounded y the curves, y =, y = & y = which lies to the right of the line =. [IIT , 5]. If c c f( ) f() = f() c + +, f() is qudrtic function nd its mimum vlue occurs t + c point V. A is point of intersection of y = f() with is nd point B is such tht chord AB sutends right ngle t V. Find the re enclosed y f() nd cheord AB. [IIT  5, 6] ANSWER EXERCISE IV. B. C. B. B 5. B 6. D 7. C 8. A 9. C. B. A. A. D. C 5. B 6. A 7. D 8. C 9. A. C. C EXERCISE V. / sq. units. c = π 6 or π.. = ln 5. (i) m =, (ii) m = ; A min = / 6. = + e, = + e ( ). 7 6 sq. units π sq. units. 5 squre units. Pge 7 of Are Under Curve
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