Use the arc length formula to find the length of the curve y = 9 - 4x, -2 s x S 3. Check your answer by noting that the curve is a line segment and calculating its length by the distance formula. 18 16 14 12 10 2 X -3 -1 1 3 -2 -4Use Simpson's Rule with n = 10 to estimate the arc length of the curve. (Round your answer to four decimal places.) y:xsin(x), OSXSZIT : Find the answer produced by a calculator or computer to compare with the previous result. (Round your answer to four decimal places.) Need Help? @ SubmltAnswer /1 Points] SCALCET9 8.1.045. MY NOTES PRACTICE ANOTHER A graphing calculator is recommended A hawk ying at 11 m/s at an altitude of 1.32 m accidentally drops its prey. The parabolic trajectory of the falling prey is described by the equation y = 132 , "_ until it hits the ground, where y ls its height above the ground and x is its horizontal distance traveled in meters. Calculate the distance traveled [in m) by the prey from the time it is dropped until the time it hits the ground. Express your answer correct to the nearest tenth of a meter. :m Solution First use are length formula. given the curve represented by the equation 4 = 9- 4x - 24 24 3 The formula for calculating the are length of the curve given by the equation y= f ( x) on the interval [a.b] b 1 = a NOW f ( x ) =g - 4x them f' ( x ) = -- 4 L - 101 IFIt(AndR = VTX dx )= 5NT7 ~ 20. 62 / - 2 - 2 Therefore the length of the curve is 5117 ~20.62 To check our answer we can also Calculate the length of the line Segment directly using the distance formula The end point of the line segment are ( -2, 9 - 4 (-2) ) = (- 2, 17 ) and ( 3.9 - 14 ( 8) ) = ( 3: - 3 ) Stood roli use the distance formula d= bre-mijn+ ( 9zy,)? 2 ( 3 - ( 2 ) ) 2 + ( - 3 - 17 ) 2 2 The length of the line segment calculated 1 425 2 5 NT7 = 20 . 62 using the distance formula matches our privious result, Thus we have confirmed own answer.solution : The curve is y = n sinn , 04 x 4 295 and n= 10 Length of the curve L Vit ( dy )2 dx Here a = 0 a b = 2r 4 = x sing dy - reosx + sink dy ) = " cost + since + 2x cosx. sink dx 275 NOW L = V it xcost + singe + 2x cosa. sink da Let & be continuous on Taib] ant in be the even integer They simpson Rule for approximationg f( a) dx is given by b ( f ( x ) d x = b - a [P ( X. ) + 4P( 21 ) + 2 R ( x2 ) + ...... + 4 f ( an ) + f(men)) where an = an-j + 2 and fx = b - a 215 - 0 n Hope f ( x ) = Vita cost + sin + 22 cosa. sinx 2 TT J R ( x ) da = \\ R (0 ) + 4 R ( * + 4x) + 2 R ( x , tax ) + ..... ... . + 4 R ( * q ) + R ( x ro ) NOW . P ( 0 ) = 1 1 + 0 + 0 + 0 = 01 F ( 24 1 ) = R ( 0 + [ ) = R ( 5 ) = 1 1 + ( # ) cos (#5 ) + sin( # )+ 2. sin($7 ) cox is similarly putting f ( x) then we solve = 1 . 482 f ( X2 ) = R ( #, + # ) = R ( 21 ) = 10671 R ( X13 ) = R ( 2n 5 + 72 ) = 8 ( 2 5 ) = 10657 R ( X 4 ) = f ( 3 5 + #5 ) = R ( 45 ) = 1. 757 f ( 25 ) = F ( 15 + 5 ) = R ( H ) = 3. 296f ( Re ) = P ( " + 5 ) = R ( em ) = 3.772 F ( x x ) = P ( 6 # + # ) = R ( 71 ) = 205 17 f ( x8 ) = R ( 7 # + $ 5 ) = R ( 8 # ) = 1. 167 R ( x q ) = R ( 8 # + 7 ) = R ( 90 ) = 4.110 R ( 240) = R ( 90 5 + 7 ) = P ( 2# ) = 6: 362 2 17 NOW | R ( x ) dx = - 1 + 4 * ( 1 . 482 ) + 2 * ( 1 . 671 ) + 4 * ( 1 0657 ) + 2 ( 1 . 7 5 7 ) + 4 ( 3 . 29 6 ) + 2 ( 3 . 7 72 ) + 4 ( 2 . 5 17 ) + 2 . ( 1 . 167 ) + 4. ( 4 . 110 ) + 6.362 = 75 [73. 97 87 = 15 . 49 401 [ a four decimal places) And :Solution given speed of hank flying = " mis altitude = 132 m The equation of parabolic trajectory 4 = 132 - 22 at 420 0 * = 66 33 we know, ds = V dont dyr do zdxy it say ,v dy = - 2x 33 From We get day 1+ ( - 22 ) 2 = do it 1089 Integrating Both side we got 6 5 = 1+ 4 x2- 1089 using formula art or de = 2 tartar + ar 2 log ox + tartar : we Calculate the value of s is 5 = 153 34 mm