선적분

$P$ 위에서 $z=\ln\frac{1}{x}=-\ln(x)$ 이다. 따라서 곡선 $P$ 를 다음과 같이 매개변수로 나타낸다.

$$\mathbf{r}(\theta)=\cos\theta\,\hat{\boldsymbol{\imath}}+\sin\theta\,\hat{\boldsymbol{\jmath}}-\ln(\cos\theta)\,\hat{\mathbf{k}}\qquad 0\le\theta\le\frac{\pi}{4}$$

그러면

$$\begin{aligned} \mathbf{r}\,'(\theta)&=-\sin\theta\,\hat{\boldsymbol{\imath}}+\cos\theta\,\hat{\boldsymbol{\jmath}}+\tan\theta\,\hat{\mathbf{k}}\\ \mathbf{F}\big(\mathbf{r}(\theta)\big)&=\cos\theta\,\hat{\boldsymbol{\imath}}+\sin\theta\,\hat{\boldsymbol{\jmath}}+\cos^3\theta\,\hat{\mathbf{k}}\\ \mathbf{F}\big(\mathbf{r}(\theta)\big)\cdot\mathbf{r}\,'(\theta)&=\sin\theta\cos^2\theta\\ \end{aligned}$$

이므로

$$\begin{aligned} \text{일} &=\int_P\mathbf{F}\cdot \mathrm{d}\mathbf{r} =\int_0^{\pi/4}\mathbf{F}\big(\mathbf{r}(\theta)\big)\cdot\mathbf{r}\,'(\theta)\ \mathrm{d}\theta =\int_0^{\pi/4}\sin\theta\cos^2\theta\ \mathrm{d}\theta =-\frac{1}{3}\cos^3\theta\Big|_0^{\pi/4} \\ &=\frac{1}{3}\big[1-\frac{1}{2^{3/2}}\big]\approx 0.2155 \end{aligned}$$