To solve this problem, we need to approximate the area under the curve $4x^2 - x - 1$ on the interval $[4, 14]$ using the left Riemann sum with 10 rectangles.

Step-by-Step Solution:

1. Width of Each Rectangle ($\Delta x$): $\Delta x = \frac{14 - 4}{10} = 1$

2. Left Endpoints: The left endpoints are: $x_0 = 4, \, x_1 = 5, \, x_2 = 6, \dots, x_9 = 13$

3. Function Values: Compute $f(x) = 4x^2 - x - 1$ at each left endpoint:

   • $f(4) = 4(4^2) - 4 - 1 = 63$
   • $f(5) = 4(5^2) - 5 - 1 = 94$
   • $f(6) = 4(6^2) - 6 - 1 = 137$
   • $f(7) = 4(7^2) - 7 - 1 = 192$
   • $f(8) = 4(8^2) - 8 - 1 = 259$
   • $f(9) = 4(9^2) - 9 - 1 = 338$
   • $f(10) = 4(10^2) - 10 - 1 = 429$
   • $f(11) = 4(11^2) - 11 - 1 = 532$
   • $f(12) = 4(12^2) - 12 - 1 = 647$
   • $f(13) = 4(13^2) - 13 - 1 = 774$

4. Approximate Area: Using the left Riemann sum: $\text{Area} = \Delta x \cdot \sum_{i=0}^{9} f(x_i)$ Substituting the values: $\text{Area} = 1 \cdot (63 + 94 + 137 + 192 + 259 + 338 + 429 + 532 + 647 + 774) = 2925$

Final Answer:

The approximate area is 2925.

Do you want a breakdown of any particular step or further explanation?

Related Questions:

1. How would the calculation change if we used right endpoints instead of left endpoints?
2. What is the integral of $4x^2 - x - 1$ over $[4, 14]$, and how does it compare to this approximation?
3. What happens to the accuracy of the approximation if we increase the number of rectangles?
4. Could this problem be solved using a midpoint Riemann sum instead?
5. What are the implications of rounding intermediate values during calculations?

Tip:

Always double-check endpoint values and computation to avoid small errors when working with Riemann sums.