Интегралы от векторных функций

Consider a drone moving through space with a velocity that varies over time. Velocity can be expressed as a vector-valued function, in which each scalar component is a real-valued function describing motion in a specific direction.

In two-dimensional space, the drone’s motion can be represented by horizontal and vertical components of the velocity function. The indefinite integral of the function is obtained by integrating each component independently, resulting in a new vector function that represents position at any time t.

In definite integration, limits are applied to each scalar component, yielding the net change in position along the x-axis and the y-axis over the specified time interval.

In three-dimensional space, an additional component accounts for motion along the third axis.

Each component of the velocity function is integrated independently. Through this process, the complete position function in three dimensions is determined.

When the vector-valued velocity function of the drone is integrated over definite limits, the result is a new vector representing the total displacement of the drone in all three spatial directions.