RI Applications of Integration Solns

RAFFLES INSTITUTION H2 Mathematics 9758 2023 Year 6 Term 3 Revision 11 (Summary and Tutorial) Topic: Applications of Integration Summary for Applications of Integration 1 Evaluating Areas Bounded by Curves and Axes 1.1 Area bounded by the curve and the x-axis In Fig. 1.1a, (i) f( ) d 0. b a xx >∫ Thus, area under f( )yx= from xa= to xb= is given by f( ) d b a xx∫ . (ii) f( ) d 0. c b xx <∫ Thus, area under f( )yx= from xb= to xc= is given by f( ) d c b xx−∫ or f( ) d c b xx∫ (see Fig. 1.1b). (iii) The area under f( )yx= from xa= to xc= is given by ( ) f( ) d f( ) d bc ab xx xx +−∫∫ or f( ) d c a xx∫ (see Fig. 1.1b). Fig 1.1a Fig 1.1b

1.2 Area bounded by the curve and the y-axis In Fig.1.2, (i) g( ) d 0. a b yy >∫ Thus, area bounded by g( )xy= and the y-axis from yb= to ya= is given by g( ) d a b yy∫ . (ii) g( ) d 0. b c yy <∫ Thus, area bounded by g( )xy= and the y-axis from yc= to yb= is given by g( ) d . b c yy−∫ (iii) The shaded area bounded by the curve g( )xy= and the y-axis from yc= to ya= is given by ( ) g( ) d g( ) d ab bc yy yy +−∫∫ . g( )xy= Fig 1.2 y x a b c

1.3 Area bounded between two curves In general, given 2 curves defined by the equations ( )1fyx= and ( )2fyx= , and an interval [ ],ab where ( ) ( )12ff xx ≥ within this interval, we can evaluate the area bounded between them from xa= to xb= by evaluating the definite integral ( ) ( )( ) 12ffd b a x xx−∫ . Similarly, given 2 curves defined by the equations ( )1gxy= and ( )2gxy= , and an interval [ ],cd where ( ) ( )12gg yy ≥ within this interval, we can evaluate the area bounded between them from yc= to yd= by evaluating the definite integral ( ) ( )( ) 12ggd d c y yy−∫ . xa b 1f( )yx= 2f()yx= x a b 1f( )yx= 2f()yx= x a b 1f( )yx= 2f()yx= x y c d 1g( )xy= 2g( )xy= x y c d 1g( )xy= 2g( )xy= x y c d 1g( )xy= 2g( )xy=

1.4 Evaluating areas involving curves which are defined pa rametrically Essentially, we write the required area as f( ) d b a xx∫ or d c g( ) dyy∫ like how we did in the earlier sections. However, since both y and x are now expressed in terms of a parameter t , the integration here is done with respect to the parameter t instead of the variable x. To do so, we apply the integration by substitution technique to change the integral to one involving the parameter t . Students are encouraged to sketch the curve with their GC in order to identify the required region. Example The parametric equations of a curve C is given by 3sin , cosx ty t= = , where 02 πt≤≤ . Find the area of bounded by the curve C, the lines x = 1 and x = 3. Solution: Required area 3 1 2 d yx= ∫ ( ) 2 1 1 3 2 1 1 3 sin sin d2 cos d d 2 cos (3cost) d xtt t tt π π − − =  = ∫ ∫