This paper investigates target detection, barrier coverage, and path coverage with randomly deployed sensors and analyzes the performance of target detection, barrier coverage, and path coverage using integral geometry. Explicit formulas of their performance are derived. The optimal convex sensing area shape with a power consumption constraint is derived from the explicit formulas. Surprisingly, the optimal convex sensing area for target detection in a convex surveillance area can be different from that for barrier coverage. A slender sensing area is optimal for the former, but a disk-shaped sensing area can be optimal for the latter. Similar results are obtained with the Boolean and probabilistic detection models. A slender sensing area is optimal for the Boolean detection model and one of the probabilistic detection models, whereas the disk-shaped sensing area is optimal for another probabilistic detection model. This paper also derives the most difficult path and target to be detected.