Approximate area under y=sin(x) using Riemann sums

1. Analyze Figure B: In Figure B, the interval $[0, \pi]$ is divided into $n=4$ subintervals, each of width $\Delta x = \frac{\pi - 0}{4} = \frac{\pi}{4}$. The rectangles are centered at $x_1 = \frac{\pi}{8}$, $x_2 = \frac{3\pi}{8}$, $x_3 = \frac{5\pi}{8}$, and $x_4 = \frac{7\pi}{8}$. The heights of these rectangles are given by $f(x) = \sin(x)$.

2. Calculate the heights:
$h_1 = \sin(\frac{\pi}{8}) \approx 0.3827$
$h_2 = \sin(\frac{3\pi}{8}) \approx 0.9239$
$h_3 = \sin(\frac{5\pi}{8}) \approx 0.9239$
$h_4 = \sin(\frac{7\pi}{8}) \approx 0.3827$

1. Calculate the total area: The area is the sum of the areas of the four rectangles: $A = \Delta x(h_1 + h_2 + h_3 + h_4) = \frac{\pi}{4}(0.3827 + 0.9239 + 0.9239 + 0.3827) = \frac{\pi}{4}(2.6132) \approx 2.0525$. Note: Depending on the specific Riemann sum method (midpoint vs right/left), the calculation varies. Based on the visual, these are midpoint rectangles. Calculating $\frac{\pi}{4} \times 2.6132 \approx 2.053$.

1. Conclusion for part (b): As the number of rectangles ($n$) increases, the width of each rectangle ($\Delta x = \frac{\pi}{n}$) decreases, and the sum of the areas of the rectangles approaches the definite integral $\int_{0}^{\pi} \sin(x) dx = 2$. Therefore, to obtain a more accurate approximation, one must continually increase the number of rectangles.