Integrals of Vector Functions

Vector-valued functions provide a convenient framework for describing motion in space when both magnitude and direction are important. A drone’s velocity at any instant has a direction and a speed, and as the drone moves, both can change. A vector-valued function captures this behavior by assigning to each time a vector whose components are real-valued functions. Each component represents motion along a particular axis in space. Such functions can describe motion in either two-dimensional or three-dimensional space, depending on the number of components included in the vector.

In two dimensions, the velocity of the drone is described by two component functions: one for horizontal motion and one for vertical motion. To find the drone’s position at any time, the velocity function is integrated with respect to time. This is done by integrating each component function separately. The result is a new vector function whose components give the horizontal and vertical positions of the drone at time t. The constant of integration appears as a constant vector, which corresponds to the drone’s initial position at the reference time. The indefinite integral represents the family of possible position functions associated with a given velocity function. Physically, it describes the position of the object at any instant in time, together with information about its initial position.

When the net change in position over a time interval is required, definite integration is used. By applying the limits of integration to each component function, the total change in horizontal and vertical position during the interval is obtained. These values form a vector that represents the drone’s net displacement in the plane. The displacement vector indicates both the direction and the distance of the overall change in position. The definite integral represents the total displacement of the object over a specified time interval. Physically, it measures the overall change in position between the initial and final times rather than the position at an individual instant.

In three-dimensional space, motion includes an additional component corresponding to the third axis. The velocity function, therefore, has three components, representing motion along each spatial direction. As in two dimensions, each component is integrated independently. The resulting position function combines all three position components and describes the drone’s motion throughout space. When evaluated over a specific time interval, the integral produces a displacement vector representing the drone’s total change in position in three dimensions.