To estimate the area under the graph of $f(x) = 5 \cos(x)$ from $x = 0$ to $x = \frac{\pi}{2}$ using four approximating rectangles with left endpoints, we proceed as follows:

1. Interval Width ($\Delta x$): $\Delta x = \frac{\frac{\pi}{2} - 0}{4} = \frac{\pi}{8}$

2. Left Endpoints: Since we are using the left endpoints, the $x$-values at these points are: $x_0 = 0, \quad x_1 = \frac{\pi}{8}, \quad x_2 = \frac{\pi}{4}, \quad x_3 = \frac{3\pi}{8}$

3. Function Values at Left Endpoints: Calculate $f(x)$ at each left endpoint: $f(x_0) = 5 \cos(0) = 5$ $f(x_1) = 5 \cos\left(\frac{\pi}{8}\right) \approx 4.6194$ $f(x_2) = 5 \cos\left(\frac{\pi}{4}\right) = 5 \times \frac{\sqrt{2}}{2} \approx 3.5355$ $f(x_3) = 5 \cos\left(\frac{3\pi}{8}\right) \approx 2.1651$

4. Estimate the Area: Now, sum the areas of the rectangles using $f(x_i) \cdot \Delta x$: $\text{Area} \approx \Delta x \times \left(f(x_0) + f(x_1) + f(x_2) + f(x_3)\right)$ Substitute the values: $\text{Area} \approx \frac{\pi}{8} \times (5 + 4.6194 + 3.5355 + 2.1651)$ $= \frac{\pi}{8} \times 15.3199 \approx 6.0165$

So, the estimated area under the curve is approximately 6.0165.

Would you like further details on any step or have questions?

Here are five related questions that could expand on this:

1. How would the estimate change if we used right endpoints instead of left endpoints?
2. What would the result be with a higher number of rectangles (e.g., 8 rectangles)?
3. How would we estimate the area if we used the midpoint instead of the left endpoint?
4. Can we calculate the exact area under the curve using integration, and how does it compare to the approximation?
5. How would the estimation process differ if the function were sinusoidal, like $f(x) = 5 \sin(x)$, instead?

Tip: Increasing the number of rectangles generally improves the accuracy of Riemann sum approximations by better capturing the curve's shape.