APPLICATION OF INTEGRALS
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1 Chpter 8 APPLICATION OF INTEGRALS 8.1 Overview This chpter dels with specific ppliction of integrls to find the re under simple curves, re etween lines nd rcs of circles, prols nd ellipses, nd finding the re ounded y the ove sid curves The re of the region ounded y the curve y f (), -is nd the lines nd ( > ) is given y the formul: Are yd f ( d ) 8.1. The re of the region ounded y the curve φ (y), y-is nd the lines y c, y d is given y the formul: d Are dy ( y) dy c d c 8.1. The re of the region enclosed etween two curves y f (), y g () nd the lines, is given y the formul. Are f ( ) g( ) d, where f () g () in [, ] 8.1. If f () g () in [, c] nd f () g () in [c, ], < c <, then c Are f ( ) g( ) d g( ) f ( ) d c 8. Solved Emples Short Answer (S.A.) Emple 1 Find the re of the curve y sin etween nd. Solution We hve Are OAB o yd sin d cos o cos cos sq units.
2 APPLICATION OF INTEGRALS 171 Emple Find the re of the region ounded y the curve y, the y-is nd the lines y nd y. Solution We hve Are BMNC 1 dy y dy 1 y 1 1 ( ) 1. 1 sq units. Emple Find the re of the region ounded y the prol y nd the stright line y. Solution The intersecting points of the given curves re otined y solving the equtions y nd y for nd y. We hve y 8 + y i.e., (y ) (y + ) which gives y, nd 8,. Thus, the points of intersection re (8, ), (, ). Hence 1 Are + y y dy y 1 y+ y 18 sq units. 6 Emple Find the re of the region ounded y the prols y 6 nd 6y.
3 17 MATHEMATICS Solution The intersecting points of the given prols re otined y solving these equtions for nd y, which re (, ) nd (6, 6). Hence Are OABC 6 6 d (6) (6) 6 1 sq units. 18 Emple Find the re enclosed y the curve cost, y sint. Solution Eliminting t s follows: cost, y sint y sin t, we otin y 1, cost, 9 which is the eqution of n ellipse. From Fig. 8., we get the required re 9 d sin Long Answer (L.A.) 6 sq units. Emple 6 Find the re of the region included etween the prol y line y + 1. Solution Solving the equtions of the given curves y we get 6 ( ) ( + ) nd the nd y + 1,
4 APPLICATION OF INTEGRALS 17, which give y 1, y From Fig.8.6, the required re re of ABC 1 d d sq units. 1 Emple 7 Find the re of the region ounded y the curves t nd y t etween the ordinte coresponding to t 1 nd t. Solution Given tht t...(i), y y t...(ii) t putting the vlue of t in (i), we get y Putting t 1 nd t in (i), we get, nd Required re re of ABCD yd d 8 ( ) 6 sq units. Emple 8 Find the re of the region ove the -is, included etween the prol y nd the circle + y. Solution Solving the given equtions of curves, we hve + or,, which give y. y ±
5 17 MATHEMATICS From Fig. 8.8 re ODAB ( ) d Let sin θ. Then d sinθ cosθ dθ nd, θ, θ. Agin, d ( sinθcos ) ( sin cos ) θ θ θ dθ sin 1 cos d θ θ θ θ Further more, ( ). d Thus the required re sq units. Emple 9 Find the re of minor segment of the circle + y cut off y the line. Solution Solving the eqution + y nd, we otin their points of intersection which re, nd,.
6 APPLICATION OF INTEGRALS 17 Hence, from Fig. 8.9, we get Required Are Are of OAB 1 + sin... 6 ` ( 6 ) 1 ( ) 1 sq units. Ojective Type Questions d Choose the correct nswer from the given four options in ech of the Emples 1 to 1. Emple 1 The re enclosed y the circle + y is equl to (A) sq units (B) sq units (C) sq units (D) sq units Solution Correct nswer is (D); since Are 1 sq. units. sin Emple 11 The re enclosed y the ellipse y + 1 is equl to (A) (B) (C) (D) Solution Correct nswer is (B); since Are d
7 176 MATHEMATICS 1 + sin. Emple 1 The re of the region ounded y the curve y nd the line y 16 (A) ` (B) 6 Solution Correct nswer is (B); since Are 16 (C) 6 Fill in the lnks in ech of the Emples 1 nd 1. ydy (D) 18 Emple 1 The re of the region ounded y the curve y, y-is nd the line y nd y is. 7 Solution sq. units Emple 1 The re of the region ounded y the curve y +, -is nd the line nd is equl to. Solution 97 sq. units 6 8. EXERCISES Short Answer (S.A.) 1. Find the re of the region ounded y the curves y 9, y.. Find the re of the region ounded y the prol y p, py.. Find the re of the region ounded y the curve y nd y + 6 nd.. Find the re of the region ounded y the curve y, y.. Find the re of the region included etween y 9 nd y 6. Find the re of the region enclosed y the prol y nd the line y + 7. Find the re of region ounded y the line nd the prol y 8 8. Sketch the region {(, ) : y } nd -is. Find the re of the region using integrtion. 9. Clculte the re under the curve y included etween the lines nd Using integrtion, find the re of the region ounded y the line y + 7, - is nd the lines nd 8.
8 APPLICATION OF INTEGRALS Drw rough sketch of the curve y 1 in the intervl [1, ]. Find the re under the curve nd etween the lines 1 nd. 1. Determine the re under the curve y included etween the lines nd. 1. Find the re of the region ounded y y nd y. 1. Find the re enclosed y the curve y nd the stright lilne + y Find the re ounded y the curve y, y + in the first qudrnt nd -is. Long Answer (L.A.) 16. Find the re of the region ounded y the curve y nd + y. 17. Find the re ounded y the curve y sin etween nd. 18. Find the re of region ounded y the tringle whose vertices re ( 1, 1), (, ) nd (, ), using integrtion. 19. Drw rough sketch of the region {(, y) : y 6 nd + y 16 }. Also find the re of the region sketched using method of integrtion.. Compute the re ounded y the lines + y, y 1 nd + y Find the re ounded y the lines y +, y nd y +.. Find the re ounded y the curve y cos nd the -is from to.. Drw rough sketch of the given curve y ,,, y nd find the re of the region ounded y them, using integrtion. Ojective Type Questions Choose the correct nswer from the given four options in ech of the Eercises to.. The re of the region ounded y the y-is, y cos nd y sin, is (A) sq units (B) ( + 1) sq units (C) ( 1) sq units (D) ( 1) sq units. The re of the region ounded y the curve y nd the stright line y is (A) 8 sq units (B) 8 sq units (C) 7 8 sq units (D) 9 8 sq units 6. The re of the region ounded y the curve y 16 nd -is is (A) 8 sq units (B) sq units (C) 16 sq units (D) 6 sq units
9 178 MATHEMATICS 7. Are of the region in the first qudrnt enclosed y the -is, the line y nd the circle + y is (A) 16 sq units (B) sq units (C) sq units (D) sq units 8. Are of the region ounded y the curve y cos etween nd is (A) sq units (B) sq units (C) sq units (D) 1 sq units 9. The re of the region ounded y prol y nd the stright line y is (A) sq units (B) 1 sq units (C) sq units (D) 1 sq units. The re of the region ounded y the curve y sin etween the ordintes, nd the -is is (A) sq units (B) sq units (C) sq units (D) 1 sq units 1. y The re of the region ounded y the ellipse + 1 is 16 (A) sq units (B) sq units (C) 16 sq units (D) sq units. The re of the region ounded y the circle + y 1 is (A) sq units (B) sq units (C) sq units (D) sq units. The re of the region ounded y the curve y + 1 nd the lines nd is (A) 7 sq units (B) 9 sq units (C) 11 sq units (D) 1 sq units. The re of the region ounded y the curve y + nd the y lines. y 1 nd y 1 is (A) sq units (B) sq units (C) 6 sq units (D) 8 sq units
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