Definite Integrals

A particle moves along a line with velocity $v(t) = 2t^3 - 3t^2$ meters per second. Find the net displacement of the particle from $t = 1$ to $t = 2$ seconds.

A particle's velocity is given by $v(t) = t^2 \sin(t)$ (in m/s). Find the net displacement of the particle from $t=0$ to $t=\pi$.

The rate at which a bacteria population grows is given by $r(t) = t e^{t^2}$ thousand bacteria per hour. What is the total increase in population (in thousands) from $t = 0$ to $t = 1$ hour? Use the Fundamental Theorem of Calculus to evaluate the definite integral.