Question

Calculus again!

Answer 1

Not. Both are diferent.

The displacement  from time t = 0 to time t = π is given by:

${\displaystyle s = \int _0^{\pi }4\cos\left(t\right)\:dt}$

${\displaystyle s = 4\left[\sin \left(t\right)\right]^{\pi }_0}$

$s = 4[0 - 0 ]$

$s = 0$

The distance from time t = 0 to time t = π is:

${\displaystyle d = \int _0^{\pi }|4\cos\left(t\right)|\:dt}$

${\displaystyle d = \int _0^{\frac{\pi }{2}}4\cos \left(t\right)dt+\int _{\frac{\pi }{2}}^{\pi }-4\cos \left(t\right)dt}$

$d = 4\left[\sin \left(t\right)\right]^{\frac{\pi}{2}}_0 - 4\left[\sin \left(t\right)\right]^{\pi}_{\frac{\pi}{2}}$

$d = 4 + 4$

$d = 8$