Question

.Determine the length of curve 𝑥 =

2

3

(𝑦 − 1)

3

2 when 𝑦 ∈ [1,4]

Answer 1

Given

$$x=\frac{2}{3}(y-1)^\frac{3}{2}$$

Length of curve is given by $L=\int_{0}^4 \sqrt{1+(\frac{dx}{dy})^2}dy$

Let us evaluate the above definite integral.

By differentiating with respect to y,

$$\frac{dx}{dy} =(y-1)^{1/2}$$

So, the integrand can be simplified as

$$\sqrt{1+(\frac{dx}{dy})^2}=\sqrt{1+[(y-1)^{1/2}]^2}=\sqrt{y}=y^{1/2}$$

We have,

$$L=\int_{0}^4y^{1/2}dy=[\frac{2}{3}y^{3/2}]_0^4 \newline=\frac{2}{3}(4)^{3/2}-2/3(0)^{3/2}\newline=16/3$$

Hence the length of the curve is $\frac{16}{3}$ .