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To estimate the total revenue using rectangles of width 10 liters with left endpoints, we need to:

1. Calculate the demand \( D(q) \) at each left endpoint of the rectangles.
2. Multiply each demand value by the width of the rectangle (10 liters).
3. Sum these products to obtain the approximate total revenue.

The given demand function is:
\[ D(q) = \frac{1}{10}q^2 - 10q + 269 \]

We are using rectangles of width 10 liters. The intervals will be:
\[ [0, 10], [10, 20], [20, 30], [30, 40], [40, 50] \]

Using the left endpoints, the values of \( q \) are: \( 0, 10, 20, 30, 40 \).

Let's calculate \( D(q) \) for each of these values:

1. \( D(0) = \frac{1}{10}(0)^2 - 10(0) + 269 = 269 \)
2. \( D(10) = \frac{1}{10}(10)^2 - 10(10) + 269 = \frac{100}{10} - 100 + 269 = 10 - 100 + 269 = 179 \)
3. \( D(20) = \frac{1}{10}(20)^2 - 10(20) + 269 = \frac{400}{10} - 200 + 269 = 40 - 200 + 269 = 109 \)
4. \( D(30) = \frac{1}{10}(30)^2 - 10(30) + 269 = \frac{900}{10} - 300 + 269 = 90 - 300 + 269 = 59 \)
5. \( D(40) = \frac{1}{10}(40)^2 - 10(40) + 269 = \frac{1600}{10} - 400 + 269 = 160 - 400 + 269 = 29 \)

Next, we multiply each of these values by the width of the rectangles (10 liters):

1. \( 269 \times 10 = 2690 \)
2. \( 179 \times 10 = 1790 \)
3. \( 109 \times 10 = 1090 \)
4. \( 59 \times 10 = 590 \)
5. \( 29 \times 10 = 290 \)

Finally, we sum these products to get the total approximate revenue:

\[ 2690 + 1790 + 1090 + 590 + 290 = 6450 \]

So, the approximate revenue using left endpoints is \( \boxed{6450} \).

Would you like more details on any step, or have any other questions?

Here are some additional questions you might have:
1. How do we calculate the total revenue using right endpoints?
2. How does the choice of endpoints (left vs. right) affect the accuracy of the approximation?
3. What is the integral that represents the exact total revenue?
4. How can we improve the accuracy of the approximation?
5. What other methods are there to estimate the area under a curve?

**Tip:** When approximating areas under curves, using more intervals with smaller widths generally increases the accuracy of your estimation.

Learn how to estimate total revenue using rectangles with a step-by-step guide. This problem involves using the rectangle method for area approximation based on a demand function. Suitable for advanced high school and undergraduate students.

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