Line Integrals - Part I — Question 4

Question 4

Let $C$ be the helix $$\mathbf r(t)=\langle 2\cos t,2\sin t,t\rangle, \qquad 0\le t\le\pi.$$ Evaluate $$\int_C(x^2+y^2+z)\,ds.$$

Tasks

1. Express the scalar function along the helix.

2. Compute the speed and evaluate the integral.

3. Verify the result by separating the constant and variable parts.

Question 4 – Solution

Strategy. Substitute the parametrization into both the scalar function and the arc-length factor.

Step 1: Restrict the integrand Along $C$, $$x^2+y^2+z =4\cos^2t+4\sin^2t+t=4+t.$$

Step 2: Arc length and integral $$\mathbf r'(t)=\langle-2\sin t,2\cos t,1\rangle, \qquad |\mathbf r'(t)|=\sqrt{4+1}=\sqrt 5.$$ Therefore $$\begin{align*} \int_C(x^2+y^2+z)\,ds &=\sqrt 5\int_0^\pi(4+t)\,dt\\ &=\sqrt 5\left[4t+\frac{t^2}{2}\right]_0^\pi =\boxed{\sqrt 5\left(4\pi+\frac{\pi^2}{2}\right)}. \end{align*}$$

Verification The constant part $4$ contributes $4L=4\pi\sqrt 5$ because $L=\pi\sqrt 5$. The $z=t$ part has average $\pi/2$ and contributes $(\pi/2)L=\pi^2\sqrt 5/2$, matching the two terms above.