Motion problems are very common throughout calculus. With derivatives, we calculated an object's velocity given its position function. With integrals, we go in the opposite direction: given the velocity function of a moving object, we find out about its position or about the change in its position. 🚘
When integrating the acceleration function, you will get the velocity function. When integrating the velocity function, you will get the position function. This concept is essential to understand when learning about the applications of integration. When you have an indefinite integral of a function, the result will be the function that comes before it. ↩️
In essence, using integrals for PVA (position, velocity, acceleration) problems is just the opposite of derivation. If you're given a velocity function and need to position, just integrate! This is because of the fundamental theorem of calculus.
"For a particle in rectilinear motion over an interval of time, the definite integral of velocity represents the particle’s displacement over the interval of time, and the definite integral of speed represents the particle’s total distance traveled over the interval of time."