$\begin{aligned} & \text { Let } I=\int_0^{50 \pi} \sqrt{1-\cos ^2 \mathrm{x}} \mathrm{dx} \\ & =\sqrt{2} \int_0^{50 \pi} \sqrt{\sin ^2 x} d x=\sqrt{2} \int_0^{50 \pi}|\sin x| d x \\ & =50 \sqrt{2} \int_0^\pi \sin x d x=-50 \sqrt{2}[\cos x]_0^\pi \\ & =100 \sqrt{2}\end{aligned}$