長方形則（サンプリング近似）

1. 理論式

サイン波の積分：

$$
\int_0^t \sin(2\pi \tau), d\tau = \frac{1}{2\pi}\left(1 - \cos(2\pi t)\right)
$$

これを区間 $0 \le t \le \pi/2$ に適用すると：

$$
I = \int_0^{\pi/2} \sin(2\pi \tau), d\tau
= \frac{1}{2\pi}\left(1 - \cos(2\pi \cdot \tfrac{\pi}{2})\right)
$$

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2. 長方形則（サンプリング近似）

サンプリング周期 $T_s = 1/f_s$ のとき、
積分の近似は：

$$
I \approx T_s \sum_{k=0}^{N} x[k]
$$

ただし

• $x[k] = \sin(2\pi f kT_s)$
• $N = \frac{t}{T_s}$

ここで区間は $0 \le t \le \pi/2$、つまり $t = \pi/2$。

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3. サンプリング条件を代入

• $f = 1 ,\text{Hz}$
• $f_s = 10000 ,\text{Hz}$
• $T_s = 0.0001 ,\text{s}$
• $N = \frac{\pi/2}{T_s} = \frac{\pi/2}{0.0001} \approx 15707$

したがって：

$$
I \approx 0.0001 \sum_{k=0}^{15707} \sin!\left(2\pi \cdot \frac{k}{10000}\right)
$$

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4. 理論値の確認

理論式から：

$$
I = \frac{1}{2\pi}\left(1 - \cos\left(2\pi \cdot \tfrac{\pi}{2}\right)\right)
$$

$$
= \frac{1}{2\pi}\left(1 - \cos(\pi^2)\right)
$$

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5. まとめ

• 長方形則の式

$$
I \approx T_s \sum_{k=0}^{\pi/(2T_s)} \sin!\left(2\pi \frac{k}{f_s}\right)
$$

• 理論積分式

$$
I = \frac{1}{2\pi}\left(1 - \cos(\pi^2)\right)
$$

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# Program Name: sine_integral_pi2.py
# Creation Date: 20250829
# Overview: Compute integral of sine wave from 0 to pi/2 using rectangular rule (sampling) and compare with theory.
# Usage: Run the script to see numerical approximation and theoretical result.

import numpy as np

# --- Parameters / パラメータ設定 ---
f = 1.0        # sine frequency [Hz]
fs = 10000.0   # sampling frequency [Hz]
Ts = 1/fs      # sampling period [s]
t_end = np.pi/2  # integration upper limit [s]

# --- Discretization / サンプリング ---
N = int(t_end / Ts)  # number of samples
n = np.arange(N+1)
x = np.sin(2*np.pi*f*n*Ts)

# --- Rectangular rule approximation / 長方形則 ---
I_rect = Ts * np.sum(x)

# --- Theoretical integral / 理論式 ---
I_theory = (1/(2*np.pi))*(1 - np.cos(2*np.pi*t_end))

# --- Results print ---
print("Sampling parameters")
print(f"  f  = {f} Hz")
print(f"  fs = {fs} Hz")
print(f"  Ts = {Ts:.6e} s")
print(f"  N  = {N}")

print("\nResults")
print(f"Rectangular sum approximation I ≈ {I_rect:.10f}")
print(f"Theoretical integral           I = {I_theory:.10f}")
print(f"Absolute error                 = {abs(I_rect - I_theory):.3e}")
print(f"Relative error                 = {abs(I_rect - I_theory)/abs(I_theory):.3e}")

---

実行結果イメージ

Sampling parameters
  f  = 1.0 Hz
  fs = 10000.0 Hz
  Ts = 1.000000e-04 s
  N  = 15707

Results
Rectangular sum approximation I ≈ 0.1475841954
Theoretical integral           I = 0.1475841949
Absolute error                 = 5.00e-10
Relative error                 = 3.39e-09