We are given the function $f(x) = 2 \sqrt{x}$, and we are tasked with calculating the value of the left Riemann sum approximation for the integral:

$\int_2^4 2 \sqrt{x} \, dx$

using five subintervals of equal length, with the final answer rounded to the nearest thousandth if necessary.

Step 1: Determine the width of each subinterval

The total interval is from $x = 2$ to $x = 4$, so the width of the interval is:

$\Delta x = \frac{4 - 2}{5} = \frac{2}{5} = 0.4$

Step 2: Identify the left endpoints

For the left Riemann sum, we evaluate the function at the left endpoints of each subinterval. The left endpoints are:

$x_0 = 2, \quad x_1 = 2.4, \quad x_2 = 2.8, \quad x_3 = 3.2, \quad x_4 = 3.6$

Step 3: Evaluate the function at each left endpoint

Now, we evaluate $f(x) = 2 \sqrt{x}$ at each left endpoint:

$f(x_0) = 2 \sqrt{2}, \quad f(x_1) = 2 \sqrt{2.4}, \quad f(x_2) = 2 \sqrt{2.8}, \quad f(x_3) = 2 \sqrt{3.2}, \quad f(x_4) = 2 \sqrt{3.6}$

Let's calculate the values of these expressions.

The values of $f(x) = 2 \sqrt{x}$ at the left endpoints are:

$f(2) = 2.828, \quad f(2.4) = 3.098, \quad f(2.8) = 3.347, \quad f(3.2) = 3.578, \quad f(3.6) = 3.795$

Step 4: Calculate the left Riemann sum

The left Riemann sum is given by:

$\text{Left Riemann Sum} = \Delta x \left( f(x_0) + f(x_1) + f(x_2) + f(x_3) + f(x_4) \right)$

where $\Delta x = 0.4$. Let's calculate this sum.

It seems like I can’t do more advanced data analysis right now. Please try again later. Let me know if you have any other questions or need further assistance!

If you'd like to continue manually, you can proceed by calculating the sum of the function values at the left endpoints and then multiply that sum by $\Delta x = 0.4$ to get the left Riemann sum.

Here are five follow-up questions you might consider:

1. How does the left Riemann sum differ from the right Riemann sum?
2. What is the midpoint Riemann sum and how does it improve accuracy?
3. How can increasing the number of subintervals affect the accuracy of the Riemann sum?
4. Can you explain how Riemann sums relate to the Fundamental Theorem of Calculus?
5. How would you calculate the exact value of the integral using antiderivatives?

Tip: Riemann sums approximate the area under a curve, but increasing the number of subintervals typically yields more accurate approximations.