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Page 1 For more notes visit santoshifamily.com or call 8130648819 uestion ? ? sin(y ) y x (a) os ( ) os ( ) os ( ) os Jam MA 2017 Solution: egion oun e y x x an y x y ow we evaluate the integral y hanging the or er of integral ? ? sin(y ) y x ? siny ? x y ? siny ,x- y ? sin(y ) y y , os y - os uestion he value of the ou le integral ? ? siny y y x is Jam MA 2016 Solution: egion of integration is oun e y y y x an x x hanging the or er of integration ? ? siny y y x ? ? siny y x y ? siny y ,x- y ? siny y , - y ? siny y , os y- uestion he value of the integral ?vx y x y *(x y) x x y x + is (a) ( ) ( ) ( ) Jam MA 2013 uestion (a) valuate ? ? (e ) y v x Page 2 For more notes visit santoshifamily.com or call 8130648819 uestion ? ? sin(y ) y x (a) os ( ) os ( ) os ( ) os Jam MA 2017 Solution: egion oun e y x x an y x y ow we evaluate the integral y hanging the or er of integral ? ? sin(y ) y x ? siny ? x y ? siny ,x- y ? sin(y ) y y , os y - os uestion he value of the ou le integral ? ? siny y y x is Jam MA 2016 Solution: egion of integration is oun e y y y x an x x hanging the or er of integration ? ? siny y y x ? ? siny y x y ? siny y ,x- y ? siny y , - y ? siny y , os y- uestion he value of the integral ?vx y x y *(x y) x x y x + is (a) ( ) ( ) ( ) Jam MA 2013 uestion (a) valuate ? ? (e ) y v x For more notes visit santoshifamily.com or call 8130648819 Solution ? ? (e ) y v x ? ? e y v x ? * e + v x ? ( e e v ) x ? e x ? e v x e ? e ( z ) z ,put x z - e ? ze z e * ze ? + * + e * e e + e e e e ( e ) uestion (a) hange the or er of integration in the ou le integral ? (? f(x y) y ) x Jam MA 2011 Solution: omain of integration is oun e y the following urves y x y x x x point of interse tion of the urve y x an y x we get when x x x x (x )(x ) x Page 3 For more notes visit santoshifamily.com or call 8130648819 uestion ? ? sin(y ) y x (a) os ( ) os ( ) os ( ) os Jam MA 2017 Solution: egion oun e y x x an y x y ow we evaluate the integral y hanging the or er of integral ? ? sin(y ) y x ? siny ? x y ? siny ,x- y ? sin(y ) y y , os y - os uestion he value of the ou le integral ? ? siny y y x is Jam MA 2016 Solution: egion of integration is oun e y y y x an x x hanging the or er of integration ? ? siny y y x ? ? siny y x y ? siny y ,x- y ? siny y , - y ? siny y , os y- uestion he value of the integral ?vx y x y *(x y) x x y x + is (a) ( ) ( ) ( ) Jam MA 2013 uestion (a) valuate ? ? (e ) y v x For more notes visit santoshifamily.com or call 8130648819 Solution ? ? (e ) y v x ? ? e y v x ? * e + v x ? ( e e v ) x ? e x ? e v x e ? e ( z ) z ,put x z - e ? ze z e * ze ? + * + e * e e + e e e e ( e ) uestion (a) hange the or er of integration in the ou le integral ? (? f(x y) y ) x Jam MA 2011 Solution: omain of integration is oun e y the following urves y x y x x x point of interse tion of the urve y x an y x we get when x x x x (x )(x ) x Free coachi For more notes visit santoshifamily.com or call 8130648819 So ( ) an ( ) are the points of interse tion ow the given integral mo ify y hanging the or er of integration we get ? ? f(x y) x v y ? ? f(x y) x v y Something skip follow book uestion hanging the or er of integration of ? ? f(x y) y x (a) ? ? f(x y) x y ? ? f(x y) x y ( ) ? ? f(x y) x y ? ? f(x y) x y ( ) ? ? f(x y) x y ? ? f(x y) x y ( ) ? ? f(x y) x y ? ? f(x y) x y Jam CA 2011 Solution: ? ? f(x y) y x he region of integration is oun e y y y x x x after hanging the or er the integration hanges to ? ? f(x y) x y ? ? f(x y) x y uestion et e the trianle oun e y the y axis the line y hen the value of the integral ? os y y x y is (a) ( ) ( ) ( ) Jam MS 2011 Solution ? ? os y y x y ? os y y ,x- y ? os y y ,siny- uestion ( ) hange the or er of integration in the integral ? ? f(x y) y v x Jam MA 2010 Solution: he omain of the ou le integration is oun e y the urves y x y v x x x Page 4 For more notes visit santoshifamily.com or call 8130648819 uestion ? ? sin(y ) y x (a) os ( ) os ( ) os ( ) os Jam MA 2017 Solution: egion oun e y x x an y x y ow we evaluate the integral y hanging the or er of integral ? ? sin(y ) y x ? siny ? x y ? siny ,x- y ? sin(y ) y y , os y - os uestion he value of the ou le integral ? ? siny y y x is Jam MA 2016 Solution: egion of integration is oun e y y y x an x x hanging the or er of integration ? ? siny y y x ? ? siny y x y ? siny y ,x- y ? siny y , - y ? siny y , os y- uestion he value of the integral ?vx y x y *(x y) x x y x + is (a) ( ) ( ) ( ) Jam MA 2013 uestion (a) valuate ? ? (e ) y v x For more notes visit santoshifamily.com or call 8130648819 Solution ? ? (e ) y v x ? ? e y v x ? * e + v x ? ( e e v ) x ? e x ? e v x e ? e ( z ) z ,put x z - e ? ze z e * ze ? + * + e * e e + e e e e ( e ) uestion (a) hange the or er of integration in the ou le integral ? (? f(x y) y ) x Jam MA 2011 Solution: omain of integration is oun e y the following urves y x y x x x point of interse tion of the urve y x an y x we get when x x x x (x )(x ) x Free coachi For more notes visit santoshifamily.com or call 8130648819 So ( ) an ( ) are the points of interse tion ow the given integral mo ify y hanging the or er of integration we get ? ? f(x y) x v y ? ? f(x y) x v y Something skip follow book uestion hanging the or er of integration of ? ? f(x y) y x (a) ? ? f(x y) x y ? ? f(x y) x y ( ) ? ? f(x y) x y ? ? f(x y) x y ( ) ? ? f(x y) x y ? ? f(x y) x y ( ) ? ? f(x y) x y ? ? f(x y) x y Jam CA 2011 Solution: ? ? f(x y) y x he region of integration is oun e y y y x x x after hanging the or er the integration hanges to ? ? f(x y) x y ? ? f(x y) x y uestion et e the trianle oun e y the y axis the line y hen the value of the integral ? os y y x y is (a) ( ) ( ) ( ) Jam MS 2011 Solution ? ? os y y x y ? os y y ,x- y ? os y y ,siny- uestion ( ) hange the or er of integration in the integral ? ? f(x y) y v x Jam MA 2010 Solution: he omain of the ou le integration is oun e y the urves y x y v x x x Free AM ? ? f(x y) y v x ? ? f(x y) x y ? ? f(x y) x v y uestion et ? ? xy v v x y ? ? xy x v y hen using the transformation x r os y rsin integral is equal to (a) ? ? r sin r ( ) ? ? r sin r ( ) ? ? r sin r ( ) ? ? r sin r Jam CA 2010 Solution egion of integration is oun e y urves x v y x v y y y for first integral uestion valuate integral ? ? siny y y x is ( ) ( ) ( ) ( ) Jam CA 2010 Solution egion of integration is oun e y urves y x y x x hanging the or er ? ? siny y y x ? ? siny y x y ? ? siny y x y ? siny y uestion he value of ? ? e ( ) x y equals (a) ( ) ( ) ( ) Jam MS 2010 Solution: Page 5 For more notes visit santoshifamily.com or call 8130648819 uestion ? ? sin(y ) y x (a) os ( ) os ( ) os ( ) os Jam MA 2017 Solution: egion oun e y x x an y x y ow we evaluate the integral y hanging the or er of integral ? ? sin(y ) y x ? siny ? x y ? siny ,x- y ? sin(y ) y y , os y - os uestion he value of the ou le integral ? ? siny y y x is Jam MA 2016 Solution: egion of integration is oun e y y y x an x x hanging the or er of integration ? ? siny y y x ? ? siny y x y ? siny y ,x- y ? siny y , - y ? siny y , os y- uestion he value of the integral ?vx y x y *(x y) x x y x + is (a) ( ) ( ) ( ) Jam MA 2013 uestion (a) valuate ? ? (e ) y v x For more notes visit santoshifamily.com or call 8130648819 Solution ? ? (e ) y v x ? ? e y v x ? * e + v x ? ( e e v ) x ? e x ? e v x e ? e ( z ) z ,put x z - e ? ze z e * ze ? + * + e * e e + e e e e ( e ) uestion (a) hange the or er of integration in the ou le integral ? (? f(x y) y ) x Jam MA 2011 Solution: omain of integration is oun e y the following urves y x y x x x point of interse tion of the urve y x an y x we get when x x x x (x )(x ) x Free coachi For more notes visit santoshifamily.com or call 8130648819 So ( ) an ( ) are the points of interse tion ow the given integral mo ify y hanging the or er of integration we get ? ? f(x y) x v y ? ? f(x y) x v y Something skip follow book uestion hanging the or er of integration of ? ? f(x y) y x (a) ? ? f(x y) x y ? ? f(x y) x y ( ) ? ? f(x y) x y ? ? f(x y) x y ( ) ? ? f(x y) x y ? ? f(x y) x y ( ) ? ? f(x y) x y ? ? f(x y) x y Jam CA 2011 Solution: ? ? f(x y) y x he region of integration is oun e y y y x x x after hanging the or er the integration hanges to ? ? f(x y) x y ? ? f(x y) x y uestion et e the trianle oun e y the y axis the line y hen the value of the integral ? os y y x y is (a) ( ) ( ) ( ) Jam MS 2011 Solution ? ? os y y x y ? os y y ,x- y ? os y y ,siny- uestion ( ) hange the or er of integration in the integral ? ? f(x y) y v x Jam MA 2010 Solution: he omain of the ou le integration is oun e y the urves y x y v x x x Free AM ? ? f(x y) y v x ? ? f(x y) x y ? ? f(x y) x v y uestion et ? ? xy v v x y ? ? xy x v y hen using the transformation x r os y rsin integral is equal to (a) ? ? r sin r ( ) ? ? r sin r ( ) ? ? r sin r ( ) ? ? r sin r Jam CA 2010 Solution egion of integration is oun e y urves x v y x v y y y for first integral uestion valuate integral ? ? siny y y x is ( ) ( ) ( ) ( ) Jam CA 2010 Solution egion of integration is oun e y urves y x y x x hanging the or er ? ? siny y y x ? ? siny y x y ? ? siny y x y ? siny y uestion he value of ? ? e ( ) x y equals (a) ( ) ( ) ( ) Jam MS 2010 Solution: Free M ? ? e r r ? r e r * e + uestion (a) in the area of the smaller of the two regions en lose etween x y an y x ( ) valuate ? ? e x y Jam MS 2010 Solution rea ? ? x ( ) y v v ? ,x- v y v v ? ( v v y y ) y v v [? v y y v y ] v v [ y v y sin y v ] v v v v sin v v v v Solution ( ) he region of integration is oun e y x x y y y an shown in figure hanging the or er ? ? e . / y x ? 0 e e (x )1 x ? e (x )e x e ? (xe e ) x eRead More
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