The problem involves determining the distance traveled by a particle with velocity $v(t) = 2t^2 + 3t$ over the interval $[a, b]$, where $a = 0$ and $b = 4$. The solution steps involve evaluating a Riemann sum and eventually taking its limit to compute the integral.

Explanation of Each Part

Part (a):

The value of $\bar{t}_k$ represents the $k$-th point of the subinterval in the partition, calculated as: $\bar{t}_k = a + \frac{2k - 1}{2n} (b-a)$ For $a = 0$ and $b = 4$, this becomes: $\bar{t}_k = \frac{2k - 1}{2n} \cdot 4 = \frac{4(2k - 1)}{2n} = \frac{4k - 2}{n}$

Part (b):

The velocity at $\bar{t}_k$, $v(\bar{t}_k)$, is substituted into the velocity function $v(t) = 2t^2 + 3t$: $v(\bar{t}_k) = v\left(\frac{4k - 2}{n}\right) = 2\left(\frac{4k - 2}{n}\right)^2 + 3\left(\frac{4k - 2}{n}\right)$

Part (c):

The summation of $v(\bar{t}_k)$ across all subintervals is: $\sum_{k=1}^n v(\bar{t}_k)$ This is an approximation of the total distance traveled over the interval.

Part (d):

The Riemann sum for the total distance traveled is: $\sum_{k=1}^n v(\bar{t}_k) \Delta t = \frac{b-a}{n} \sum_{k=1}^n v\left(a + \frac{2k - 1}{2n} (b-a)\right)$ For $a = 0, b = 4$, $\Delta t = \frac{4-0}{n} = \frac{4}{n}$.

Part (e):

Taking the limit as $n \to \infty$ gives the exact distance traveled: $\lim_{n \to \infty} \sum_{k=1}^n v(\bar{t}_k) \Delta t = \int_a^b v(t) \, dt$ This integral can be directly calculated: $\int_0^4 (2t^2 + 3t) \, dt$ $= \left[\frac{2t^3}{3} + \frac{3t^2}{2}\right]_0^4$

Step-by-Step Calculation of the Integral

1. Compute the antiderivative: $\frac{2t^3}{3} + \frac{3t^2}{2}$
2. Evaluate at $t = 4$ and $t = 0$: $\left(\frac{2(4)^3}{3} + \frac{3(4)^2}{2}\right) - \left(\frac{2(0)^3}{3} + \frac{3(0)^2}{2}\right)$
3. Simplify: $\frac{2(64)}{3} + \frac{3(16)}{2} = \frac{128}{3} + \frac{48}{2} = \frac{128}{3} + 24 = \frac{200}{3}$

The total distance traveled is: $\frac{200}{3} \text{ units.}$

Would you like further breakdowns or additional help? Here are related questions:

1. What is the significance of using Riemann sums in evaluating integrals?
2. How can you approximate the integral for a specific $n$?
3. What is the geometrical meaning of $v(t)$ over time $t$?
4. Can you calculate the distance using midpoint or trapezoidal rules instead?
5. How do you interpret the integral of velocity in physical terms?

Tip: Always verify your calculations by comparing numerical approximations to symbolic solutions when integrating complex functions.