Answer to Question #184938 in Civil and Environmental Engineering for John Prats

Question #184938

Determine the arc lengths defined by the following functions over the given intervals: a.) y=1/3 x^3/2 on (0,4)

b.) r= sin² (theta)/2 on 0≤(theta)≤π.


1
Expert's answer
2021-04-29T07:58:04-0400

(a.) y=1/3 x^3/2 [0,4]

dy/dx=d/dx[1/3(x)3/2

dy/dx= [1/3(x)3/2-1(3/2)]. d/dx(x+1)

dy/dx= [(x+1)1/2. (2x)

dy/dx= 2x(x+1)1/2

[a, b]= [0,4]

arc length= "\\int"40 "\\intop"1+(dy/dx)2dx

= "\\int"40 "\\int"1+(2x(x+1)1/2)dx

= "\\int"40 "\\int"1+(2x)2((x2+1)2/2)dx

= "\\int"40 "\\int"1+4x2(x2)+4x2(1)dx

= "\\int"40 "\\int"1+4x4+4x2+1dx

= "\\int"40 "\\int"4x4+4x2+1dx

= "\\int"40 "\\int"(2x2+1)2dx

= [ "\\int"(2x2dx+ "\\int"1dx]40

=[2(x3/3)+ (x)]40

=[2(4)3/3+1]-[2(0)3/3+(0)]

=[128/3+1]-[0+0]

=43.667








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