To approximate the area under the curve using the Left Endpoint approximation with 3 subdivisions, we can follow these steps:

1. Determine the interval width:

The interval is from $x = 3$ to $x = 6$. This gives us a total interval length of:

$\text{Interval length} = 6 - 3 = 3$

Since we are using 3 subdivisions, the width of each subinterval ($\Delta x$) is:

$\Delta x = \frac{3}{3} = 1$

2. Identify the left endpoints of each subinterval:

Since we're using left endpoints, the points at which we'll evaluate the function are the leftmost points of each interval:

• $x_0 = 3$
• $x_1 = 4$
• $x_2 = 5$

3. Evaluate the function at these left endpoints:

From the graph, we need to estimate the function's values at these points:

• $f(3) \approx 2$
• $f(4) \approx 3$
• $f(5) \approx 3.8$

These values are approximated based on the graph.

4. Calculate the area of each rectangle:

For each subinterval, the area of the rectangle is given by the height of the function at the left endpoint multiplied by the width $\Delta x = 1$.

$\text{Area} = f(x_i) \times \Delta x$

• For $x_0 = 3$, $\text{Area}_1 = 2 \times 1 = 2$
• For $x_1 = 4$, $\text{Area}_2 = 3 \times 1 = 3$
• For $x_2 = 5$, $\text{Area}_3 = 3.8 \times 1 = 3.8$

5. Sum the areas to approximate the total area under the curve:

Adding up the areas of the rectangles gives the total approximate area:

$\text{Total Area} = 2 + 3 + 3.8 = 8.8 \text{ square units}$

Conclusion:

The approximate area under the curve from $x = 3$ to $x = 6$ using the Left Endpoint approximation with 3 subdivisions is 8.8 square units.

Questions:

1. Would you like to explore using a different number of subdivisions?
2. Should we compare this approximation with a right-endpoint or midpoint approximation?
3. Would you like to calculate the exact area using integration?
4. Should we try a different function or interval for practice?
5. Are you interested in the concept of the limit as the number of subdivisions increases?
6. Would you like to understand how to estimate function values more accurately from graphs?
7. Do you want to see how the error in this approximation can be estimated?
8. Would you like to learn about other numerical methods like the trapezoidal rule?

Tip:

When approximating areas under a curve, increasing the number of subdivisions generally improves the accuracy of your estimate. Always be aware of the method you are using (left, right, midpoint) as it can significantly affect the result.