Question

Suppose a certain object moves in a straight line with the following velocity, where v is in meters per second and t is in seconds: v(t)=-2+t+3sin(πt).
Without using a calculator, but instead using the properties of definite integrals and facts you know about area, determine the net change in distance of the object from time t=0 to time t=6 seconds and find the object's average velocity on this interval.

Answer 1

The net change in distance is given by the displacement, which is simply the integration of the velocity function over the given interval,
\[\int_0^6v(t)\,dt\]
The average velocity on this interval can be computed via the mean value theorem, which says
\[\text{average}=\frac{1}{6-0}\int_0^6v(t)\,dt\]