A particle moves along a number line with velocity at time $t$ given by the formula $v(t) = t \operatorname{ln}(t + 1)$.

1. Use a five-term midpoint Riemann sum to estimate the displacement of the particle from $t = 0$ to $t = 4$.

   Hint (a):

   General setup for a midpoint Riemann sum:

   $\displaystyle\sum_{i=0}^{n-1}\Delta xf(a+\frac{\Delta x}{2}i)$

   Step (a):

   $\displaystyle\sum _ { i = 0 } ^ { 4 } \frac { 4 } { 5 } \Big( \frac { 2 } { 5 } + \frac { 4 i } { 5 } \Big) \operatorname { ln } \Big( \frac { 7 } { 5 } + \frac { 4 i } { 5 } \Big)$

2. Estimate the same displacement using the Trapezoid Rule with five trapezoids.

   Hint (b):

   General setup for Trapezoidal sum on $[a, b]$:

   $\frac{\Delta x}{2}[f(a)+2f(a+\Delta x)+2f(a+2\Delta x)+...+2f(b-\Delta x)+f(b)]$

   Step (b):

   $\displaystyle\sum _ { i = 0 } ^ { 4 } \Big( \frac { 4 } { 5 } \Big) \frac { v ( 4 i / 5 ) + v ( ( 4 + 4 i ) / 5 ) } { 2 }$