Question

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.

Answer 1

(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