Line Integrals - Part II — Question 8

Question 8 – Solution

Strategy. Treat each coordinate differential separately, then combine all three ordinary integrals.

Step 1: Differential data $$x=\cos t,\quad y=\sin t,\quad z=t,$$ $$dx=-\sin t\,dt,\quad dy=\cos t\,dt,\quad dz=dt.$$

Step 2: Evaluate $$\begin{align*} \int_C z\,dx+x\,dy+y\,dz &=\int_0^{\pi/2} \left(-t\sin t+\cos^2t+\sin t\right)dt. \end{align*}$$ The three contributions are $$\int_0^{\pi/2}-t\sin t\,dt=-1,\quad \int_0^{\pi/2}\cos^2t\,dt=\frac{\pi}{4},\quad \int_0^{\pi/2}\sin t\,dt=1.$$ Hence $$\boxed{\int_C z\,dx+x\,dy+y\,dz=\frac{\pi}{4}}.$$

Verification The first and third contributions cancel exactly, leaving only $\int_0^{\pi/2}\cos^2t\,dt=\pi/4$.