5. (27 pts) Let f(x) = x + 2 and g(x) = (x -4)2. A) Find the points of intersection (x and y coordinates) of f(x) and g(x).5 ( @ ) f ( x ) = 4 + 2 and gra) = ( 4 - 4 ) 2 f ( 2 ) = 9 ( 2 ) 4 +2 - ( 9- 4 ) - Now solike the squ 4+2 - 9 8 x +16 sub a+ 2 bothsides 0 = 2- 92 +14 wow factor the quadratic &g Cox - 7) ( 21 - 2 ) = 0 this gilles us two possible values of x 2- 7 =0 =90=4 x - 2 = 0 =) 2:2 for * = 4 +( 7 ) = 7+2 = 9 9 ( 7 ) = ( 7 - 4 ) ? = 32 - 9 for 21 = 2 + ( 2 ) 2 2 + 2 -4 g le ) = ( 2 - 4 )r = (-2)1 = 4 so the points of interaction age ( 7 / 9 ) ( 2 , 4 )(5 1 ) the agea f(al and grow ) 17 1 = 71 [ 3 1 ( x+5)- ( x - 4 ) dx Area : Self() - 90) kda Capand 1/ 0 - 2 and x - 1 (+2 ) - ( 2 - 4 ) " , ( 2 " + 40+4 ) - Dolegral Area ( + ( 2 + 2 ) ( 9 - 4 )" ld2) ( at - 8 x + 16 ) = 1,27 - 12 lalbere 94 +2 - ( 1 -4 ) 1 20 Now sub in flegral Area = 17 (( at 2 ) - ( x -4 )" )d \\= TT ( ? ( 124 -12)1 x Where9 42 - ( 9 - 4 ) 20 Alow integrate with respect toy Area - 13 ( (x - 4 ) 2 - (x telldy V= TV [ 6 1 ) xx - 12 x ]] Alow solide V= TI [3x k - 12 x 7 7 Area, = 1 ( (+ 2 ) - ( 1 - 4 ) ")odr V= IT [ 3 ( 7 ) 2 - 12( 7 ) - 3( 2) 1- 12.27 = 13 ( 2 + 2 - Car - 8 x + 16 )bly V= 1 [ 147 - 84-(12-24)] = 13 ( -at + 9x - 14 )ely V = 1 - 51 Areaq = 15 ( 2 -4 )2 - (x+2 )did v = 5171 =1 ( x - q2 + 14 )dy Expand Area = Arca a , Arcaas ( +2) - (2-4)7= (2+2) - (91- 8x416) Arcaa [- 1/3 23 + 91,24 -1427 7, =-95+92-1+ = [- 1/3 (7)3+ 9 ( 4)2- 14.71 -1-/ Sub 90 integral v=on [.x(27 97+ ( 2 ) 3+ 9/2 ( 2)7 - 14-27 distrublea van ! (x 3 +qu Integrate with I = [ - 343 +441 - 98 + 8/3 -1819 8 Arraq = [16 23 - 9/ () +lan]? Vani [ -/ 2 4 + 3 2 3 +x ] V= 271 - 1/4( +) *+ 3(7)- 7 ( 712 3 [343 - 441 + 98 -8/ +18-281 2 ( /4 ) ( ) + + 3 ( 2 ) 3 - 7 ( 2 ) Area: Area a - Areaz V= 271 -2401 +9261 _243 Arta 20 . *1 /4 +24-28] V= 21 6481 - 341] V= 1 [ 13792 -(94 ] V= TI [ 27 36 -5 ]] therefore the Gart Value of volume & solid formed when the region about y - aurs is 273671iv ) to find the dal & solid formed from is rotated about line 4 = - 3 V = T (7 ( x +2 ) - ( x - 4 ) ~ + 32 ]dy lets sub Exp (- 2 2+97 -11) back into Pokegral VETT ? ( - 2 + + 9 x - 11 ) dx to alaluale I used a symbolic cal using INolfarm Alpha 4 a V- 7577 -02 the vol of the solid formed if the region 2= 10 V = aT1 7 ( 10 - 2 ) [catz ) - ( x - 4 ) " ]dy to simplify the inside polegraf (10 - 2 ) [ (x + 2) - (2-4 )7 = ( 10-2) (-9 1+92-14) sub back in integral V=2TI B ( 10-2 ) (-x + +9x -14)d x let cilatuat used a symbolic calculator using wolfaim Alpha V2 46 06 - 25