The question asks to sketch rectangles for the approximation $L_4$ of the definite integral $\int_0^8 f(x)dx$. This indicates using Left Riemann Sum with 4 subintervals.

Here's the process to construct the rectangles:

1. Determine the interval width ($\Delta x$):

   • The total interval is from $x = 0$ to $x = 8$, so the width of the interval is: $\Delta x = \frac{8 - 0}{4} = 2.$
   • The subintervals are $[0, 2], [2, 4], [4, 6], [6, 8]$.

2. Use the left endpoints to calculate the rectangle heights:

   • For $L_4$, the height of each rectangle is the value of $f(x)$ at the left endpoint of each subinterval:
     • Height for $[0, 2]$: $f(0)$,
     • Height for $[2, 4]$: $f(2)$,
     • Height for $[4, 6]$: $f(4)$,
     • Height for $[6, 8]$: $f(6)$.

3. Sketch the rectangles:

   • For each subinterval, draw a rectangle starting at the left endpoint of the subinterval with a width of $\Delta x = 2$ and a height corresponding to $f(x)$ at the left endpoint.
   • The top-left corner of each rectangle touches the curve $f(x)$.

Would you like me to provide a detailed diagram of these rectangles or guide you through more calculations? Let me know!

Here are 5 related questions to expand your understanding:

1. What is the formula for the Left Riemann Sum approximation $L_n$ for a general function $f(x)$?
2. How would the result change if the Right Riemann Sum ($R_4$) was used instead?
3. How can you use a midpoint approximation for the same problem?
4. What is the interpretation of Riemann Sums as the number of intervals ($n$) approaches infinity?
5. How does $L_4$ compare to the exact integral value in this case?

Tip: Practice sketching Riemann sums for different functions and intervals to gain better intuition!