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.

Expert's answer

(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"^{4}_{0 }"\\intop"1+(dy/dx)^{2}dx

= "\\int"^{4}_{0 } "\\int"1+(2x(x+1)^{1/2})dx

= "\\int"^{4}_{0 } "\\int"1+(2x)^{2}((x^{2}+1)^{2/2})dx

= "\\int"^{4}_{0 } "\\int"1+4x^{2}(x^{2})+4x^{2}(1)dx

= "\\int"^{4}_{0 } "\\int"1+4x^{4}+4x^{2+}1dx

= "\\int"^{4}_{0 } "\\int"4x^{4}+4x^{2}+1dx

= "\\int"^{4}_{0 } "\\int"(2x^{2}+1)^{2}dx

= [ "\\int"(2x^{2}dx+ "\\int"1dx]^{4}_{0}

=[2(x^{3}/3)+ (x)]^{4}_{0}

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

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

=43.667

Learn more about our help with Assignments: Engineering

## Comments

## Leave a comment