find the arc length of the following curve on the given interval. y = 1/3 x^3/2 on 0,53 the length of the…

Explanation:

Step1: Find the derivative of y

The function is $y = \frac{1}{3}x^{\frac{3}{2}}$. Using the power - rule $(x^n)'=nx^{n - 1}$, we have $y'=\frac{1}{3}\times\frac{3}{2}x^{\frac{3}{2}-1}=\frac{1}{2}x^{\frac{1}{2}}$.

Step2: Use the arc - length formula

The arc - length formula for a function $y = f(x)$ on the interval $[a,b]$ is $L=\int_{a}^{b}\sqrt{1+(y')^{2}}dx$. Here, $a = 0$, $b = 53$, and $(y')^{2}=\frac{1}{4}x$. So, $1+(y')^{2}=1+\frac{1}{4}x=\frac{4 + x}{4}$, and $\sqrt{1+(y')^{2}}=\frac{\sqrt{4 + x}}{2}$.

Step3: Calculate the integral

$L=\int_{0}^{53}\frac{\sqrt{4 + x}}{2}dx$. Let $u=4 + x$, then $du=dx$. When $x = 0$, $u = 4$; when $x = 53$, $u = 57$. The integral becomes $\frac{1}{2}\int_{4}^{57}u^{\frac{1}{2}}du$. Using the power - rule for integration $\int x^n dx=\frac{x^{n + 1}}{n+1}+C(n\neq - 1)$, we have $\frac{1}{2}\times\frac{2}{3}u^{\frac{3}{2}}\big|{4}^{57}=\frac{1}{3}(u^{\frac{3}{2}})\big|{4}^{57}$.

Step4: Evaluate the definite integral

$\frac{1}{3}(57^{\frac{3}{2}}-4^{\frac{3}{2}})=\frac{1}{3}(57\sqrt{57}-8)$. $57\sqrt{57}\approx57\times7.55=430.35$. Then $\frac{1}{3}(430.35 - 8)=\frac{1}{3}\times422.35\approx140.78$.

Answer:

$140.78$