vertical slice formed by f (x) − g(x), and then plugging it into the appropriate area formula. In
the case of a circle, f (x) − g(x) gives us the length of the diameter and we use the following
formula:
A(x) =
This gives us the integral,
Expand the integrand.
Evaluate the integral.
- An object moves with velocity v(t) = t^2 − 8t + 7.
(a) Write a polynomial expression for the position of the particle at any time t ≥ 0.
The velocity of an object is the derivative of its position with respect to time. Thus, if we want
to find the position, we take the integral of velocity with respect to time.
(b) At what time(s) is the particle changing direction?
If we want to find when the particle is changing direction, we need to find where the velocity
of the particle is zero.
v(t) = t^2 − 8t + 7 = (t − 1)(t − 7) = 0
Thus, at t = 1 or t = 7, the particle could be changing direction. To make sure, we need to check
that the acceleration of the particle is not zero at those times. The acceleration of a particle is
