Line Integrals - Part I — Question 7

Question 7

A wire follows the first-quadrant astroid arc $$\mathbf r(t)=\langle\cos^3t,\sin^3t\rangle, \qquad 0\le t\le\frac{\pi}{2},$$ and has linear density $\delta(x,y)=x+y$. Find its mass.

Tasks

1. Derive and simplify the speed on the given interval.

2. Express the density in terms of $t$ and evaluate the mass.

3. Check positivity and endpoint behavior.

Question 7 – Solution

Strategy. Differentiate the astroid parametrization; in the first quadrant the absolute values in the speed simplify.

Step 1: Speed

$$\mathbf r'(t)= \langle-3\cos^2t\sin t,\,3\sin^2t\cos t\rangle.$$ Because $\sin t,\cos t\ge 0$, $$\begin{align*} |\mathbf r'(t)| &=3\sin t\cos t\sqrt{\cos^2t+\sin^2t}\\ &=3\sin t\cos t. \end{align*}$$

Step 2: Mass Along the wire, $\delta=\cos^3t+\sin^3t$. Hence $$\begin{align*} M&=\int_0^{\pi/2} 3(\cos^3t+\sin^3t)\sin t\cos t\,dt\\ &=3\left(\int_0^{\pi/2}\cos^4t\sin t\,dt +\int_0^{\pi/2}\sin^4t\cos t\,dt\right)\\ &=3\left(\frac 15+\frac 15\right)=\boxed{\frac 65}. \end{align*}$$

Verification Density and speed are nonnegative. The speed vanishes only at the two cusps, which does not affect the integral; the finite positive mass $6/5$ is therefore consistent.