Find the length of the spiral r 2 theta 2 0 theta The leng

Find the length of the spiral r = 2 theta 2, 0 theta . The length of the spiral is . (Simplify your answer.)

Solution

r=2t^2 , 0 r^2 = 4t^2 AND r \' = 4t L = \\int_{0}^{sqrt(21)} sqrt(r^2 + (r \' )^2 ) dt L = \\int_{0}^{sqrt(21)} sqrt(4 t^4 + 16 t^2 ) dt L = \\int_{0}^{sqrt(21)} 2 t sqrt( t^2 + 4 ) dt TAKE t^2+4 = u ==> 2t dt = du t=0==> u= 4 t=sqrt(21) ==> u= 21+4 =5 L = \\int_{4}^{25} sqrt( u ) du L = \\int_{4}^{25} u^(1/2) du = 78 ANSWER