Question

Assume the reader understands derivatives, and knows the definition of instantaneous velocity (dx/dt), and knows how to calculate integrals but is struggling to understand them. Use students’ prior knowledge to provide an explanation that includes the concept and physical meaning of the integral of velocity with respect to time.

Reminder: The user is comfortable with the calculations, but is struggling with the concept. To fully address the prompt, emphasize the written explanation in English over the calculation.

Do not want hand written answer and do not copy paste. Please type. Thanks.

Answer 1

As the reader understand derivative. In this case it is rate change of position in time.

As reader comfortable with calculas so simply integration is nothing but a area. ie, total area covered by the curve. And definite integral is area covered by two point under the curve. ie, total area covered by the curve between two points.  

Here the curve is position​​​​​​. Derivative of position with time gives the rate of change in position. Which is velocity. Now take velocity as a new curve which is actually rate of change of position.

Integral of velocity means area under by velocity curve. Which is actually total distance covered by the moving particle. ie, total displacement of moving object.

Physically how much distance will cover with velocity (dx/dt) is given by the integration with respect to time. And integration of velocity within time limit (finite integral) gives total distance covered with the velocity (dx/dt) between given two times.

Here integration of velocity is nothing but the distance covered by the moving particle.

Hope it will be clear.

(It is much easier to understand with math. )

(N.B.- velocity takes both +ve and -ve so there may be change is 0 when we calculate the distance covered with integral. Different for speed)