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<div class="md-content" data-md-component="content">
<article class="md-content__inner md-typeset">
<h1 id="_1">积分学<a class="headerlink" href="#_1" title="Permanent link">&para;</a></h1>
<p><em>积分学在区间上累积量，将局部变化率还原为总量。本文涵盖定积分与不定积分、微积分基本定理、积分技巧，以及在机器学习中与概率密度和期望值的应用。</em></p>
<ul>
<li>
<p>微分告诉我们单个点的变化率。积分则相反：它将许多微小片段累积起来，计算出一个总量。</p>
</li>
<li>
<p>如果导数回答的是"多快？"，那么积分回答的是"多少？"</p>
</li>
<li>
<p>理解积分最简单的方式是将其视为<strong>曲线下的面积</strong>。如果绘制出函数 <span class="arithmatex">\(f(x)\)</span> 的图像，并将从 <span class="arithmatex">\(x = a\)</span> 到 <span class="arithmatex">\(x = b\)</span> 之间曲线与 x 轴之间的区域涂上阴影，积分给出的就是该区域的有符号面积。</p>
</li>
</ul>
<p><img alt="积分通过求和薄矩形来计算曲线下的面积" src="../../images/area_under_curve.svg" /></p>
<ul>
<li>
<p>为什么是"有符号"的？在 x 轴上方的区域贡献正面积，在下方的区域贡献负面积。这在物理上是有意义的：如果 <span class="arithmatex">\(f(x)\)</span> 代表速度，积分给出的是净位移（正向减去反向），而不是总路程。</p>
</li>
<li>
<p>为了计算这个面积，想象将区域切成 <span class="arithmatex">\(n\)</span> 个细长的竖直矩形，每个矩形的宽度为 <span class="arithmatex">\(\Delta x\)</span>。每个矩形的高度是该切片内某一点的函数值。将它们求和：</p>
</li>
</ul>
<div class="arithmatex">\[\text{面积} \approx \sum_{i=1}^{n} f(x_i^\ast) \, \Delta x\]</div>
<ul>
<li>当我们让矩形越来越薄时（<span class="arithmatex">\(n \to \infty\)</span>，<span class="arithmatex">\(\Delta x \to 0\)</span>），这个和就变得精确。这个极限过程定义了<strong>定积分</strong>：</li>
</ul>
<div class="arithmatex">\[\int_a^b f(x)\, dx = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i^\ast) \, \Delta x\]</div>
<ul>
<li>
<p><span class="arithmatex">\(\int\)</span> 符号是拉长的"S"，代表"求和"（Sum）。<span class="arithmatex">\(dx\)</span> 提醒我们，我们是在沿 x 轴方向对无穷薄的切片求和。</p>
</li>
<li>
<p><strong>不定积分</strong>（或<strong>原函数</strong>）是一个函数 <span class="arithmatex">\(F(x)\)</span>，其导数为 <span class="arithmatex">\(f(x)\)</span>。我们写作：</p>
</li>
</ul>
<div class="arithmatex">\[\int f(x)\, dx = F(x) + C\]</div>
<ul>
<li>
<p><span class="arithmatex">\(+ C\)</span> 是<strong>积分常数</strong>。因为任何常数的导数都是零，所以存在无穷多个仅相差一个常数的原函数。例如，<span class="arithmatex">\(\int 2x\, dx = x^2 + C\)</span>，因为 <span class="arithmatex">\(x^2 + 7\)</span> 或 <span class="arithmatex">\(x^2 - 3\)</span> 的导数仍然是 <span class="arithmatex">\(2x\)</span>。</p>
</li>
<li>
<p><strong>微积分基本定理</strong>是连接微分与积分的桥梁。它包含两部分：</p>
</li>
<li>
<p><strong>第一部分</strong>：如果 <span class="arithmatex">\(F(x)\)</span> 是 <span class="arithmatex">\(f(x)\)</span> 的一个原函数，那么定积分等于 <span class="arithmatex">\(F\)</span> 在端点处的值之差：</p>
</li>
</ul>
<div class="arithmatex">\[\int_a^b f(x)\, dx = F(b) - F(a)\]</div>
<ul>
<li>
<p>这非常实用。我们不再需要计算一个和的极限（这很困难），而是找到一个原函数并在两点处求值（这通常很简单）。</p>
</li>
<li>
<p><strong>第二部分</strong>：如果我们定义 <span class="arithmatex">\(F(x) = \int_a^x f(t)\, dt\)</span>，那么 <span class="arithmatex">\(F'(x) = f(x)\)</span>。微分与积分是互逆运算，它们相互抵消。</p>
</li>
<li>
<p>例如，计算 <span class="arithmatex">\(\int_1^3 x^2\, dx\)</span>：<span class="arithmatex">\(x^2\)</span> 的原函数是 <span class="arithmatex">\(\frac{x^3}{3}\)</span>。所以 <span class="arithmatex">\(\int_1^3 x^2\, dx = \frac{27}{3} - \frac{1}{3} = \frac{26}{3} \approx 8.67\)</span>。</p>
</li>
<li>
<p>正如微分有运算法则一样，积分也有相应的逆向运算法则：</p>
</li>
</ul>
<table>
<thead>
<tr>
<th>函数</th>
<th>积分</th>
<th>条件</th>
</tr>
</thead>
<tbody>
<tr>
<td><span class="arithmatex">\(x^n\)</span></td>
<td><span class="arithmatex">\(\frac{x^{n+1}}{n+1} + C\)</span></td>
<td><span class="arithmatex">\(n \neq -1\)</span></td>
</tr>
<tr>
<td><span class="arithmatex">\(\frac{1}{x}\)</span></td>
<td><span class="arithmatex">\(\ln\|x\| + C\)</span></td>
<td></td>
</tr>
<tr>
<td><span class="arithmatex">\(e^x\)</span></td>
<td><span class="arithmatex">\(e^x + C\)</span></td>
<td></td>
</tr>
<tr>
<td><span class="arithmatex">\(a^x\)</span></td>
<td><span class="arithmatex">\(\frac{a^x}{\ln a} + C\)</span></td>
<td></td>
</tr>
<tr>
<td><span class="arithmatex">\(\sin x\)</span></td>
<td><span class="arithmatex">\(-\cos x + C\)</span></td>
<td></td>
</tr>
<tr>
<td><span class="arithmatex">\(\cos x\)</span></td>
<td><span class="arithmatex">\(\sin x + C\)</span></td>
<td></td>
</tr>
<tr>
<td><span class="arithmatex">\(k\)</span>（常数）</td>
<td><span class="arithmatex">\(kx + C\)</span></td>
<td></td>
</tr>
</tbody>
</table>
<ul>
<li>
<p><strong>和/差法则</strong>同样适用：<span class="arithmatex">\(\int [f(x) \pm g(x)]\, dx = \int f(x)\, dx \pm \int g(x)\, dx\)</span>。常数可以提出来：<span class="arithmatex">\(\int k\, f(x)\, dx = k \int f(x)\, dx\)</span>。</p>
</li>
<li>
<p>当一个函数太复杂而无法直接积分时，我们有简化它的技巧。</p>
</li>
<li>
<p><strong>换元积分法（u 代换）</strong>是链式法则的逆过程。如果发现一个复合函数 <span class="arithmatex">\(f(g(x))\)</span> 乘以 <span class="arithmatex">\(g'(x)\)</span>，则令 <span class="arithmatex">\(u = g(x)\)</span>，于是 <span class="arithmatex">\(du = g'(x)\, dx\)</span>，积分得以简化。</p>
</li>
<li>
<p>例如：<span class="arithmatex">\(\int 2x \cos(x^2)\, dx\)</span>。令 <span class="arithmatex">\(u = x^2\)</span>，则 <span class="arithmatex">\(du = 2x\, dx\)</span>。积分变为 <span class="arithmatex">\(\int \cos(u)\, du = \sin(u) + C = \sin(x^2) + C\)</span>。</p>
</li>
<li>
<p><strong>分部积分法</strong>是乘积法则的逆过程。如果被积函数是两个函数的乘积：</p>
</li>
</ul>
<div class="arithmatex">\[\int u\, dv = uv - \int v\, du\]</div>
<ul>
<li>
<p>策略性地选择 <span class="arithmatex">\(u\)</span> 和 <span class="arithmatex">\(dv\)</span>，使得剩下的积分 <span class="arithmatex">\(\int v\, du\)</span> 比原来的更简单。选择 <span class="arithmatex">\(u\)</span> 的常用助记法是 <strong>LIATE</strong>：对数函数（Logarithmic）、反三角函数（Inverse trig）、代数函数（Algebraic）、三角函数（Trigonometric）、指数函数（Exponential）（优先从靠前的类别中选择 <span class="arithmatex">\(u\)</span>）。</p>
</li>
<li>
<p>例如：<span class="arithmatex">\(\int x\, e^x\, dx\)</span>。令 <span class="arithmatex">\(u = x\)</span>（代数函数）和 <span class="arithmatex">\(dv = e^x\, dx\)</span>。则 <span class="arithmatex">\(du = dx\)</span>，<span class="arithmatex">\(v = e^x\)</span>。因此：<span class="arithmatex">\(\int x\, e^x\, dx = x\, e^x - \int e^x\, dx = x\, e^x - e^x + C = e^x(x - 1) + C\)</span>。</p>
</li>
<li>
<p>在机器学习中，积分出现在概率论中（通过对密度函数积分来计算概率）、期望值中（连续分布上的加权平均），以及计算 ROC 曲线下的面积。虽然在实际中我们很少手动积分，但理解积分的含义有助于解释这些量。</p>
</li>
</ul>
<h2 id="colab-notebook">编程练习（使用 CoLab 或 notebook）<a class="headerlink" href="#colab-notebook" title="Permanent link">&para;</a></h2>
<ol>
<li>
<p>使用黎曼和，用不断增加数量的矩形来数值逼近 <span class="arithmatex">\(\int_0^1 x^2\, dx\)</span>。与精确答案 <span class="arithmatex">\(\frac{1}{3}\)</span> 进行比较。
<div class="highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="kn">import</span><span class="w"> </span><span class="nn">jax.numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">jnp</span>
<a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a>
<a id="__codelineno-0-3" name="__codelineno-0-3" href="#__codelineno-0-3"></a><span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="p">[</span><span class="mi">10</span><span class="p">,</span> <span class="mi">100</span><span class="p">,</span> <span class="mi">1000</span><span class="p">,</span> <span class="mi">10000</span><span class="p">]:</span>
<a id="__codelineno-0-4" name="__codelineno-0-4" href="#__codelineno-0-4"></a> <span class="n">x</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">endpoint</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
<a id="__codelineno-0-5" name="__codelineno-0-5" href="#__codelineno-0-5"></a> <span class="n">dx</span> <span class="o">=</span> <span class="mf">1.0</span> <span class="o">/</span> <span class="n">n</span>
<a id="__codelineno-0-6" name="__codelineno-0-6" href="#__codelineno-0-6"></a> <span class="n">area</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span> <span class="o">*</span> <span class="n">dx</span><span class="p">)</span>
<a id="__codelineno-0-7" name="__codelineno-0-7" href="#__codelineno-0-7"></a> <span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;n=</span><span class="si">{</span><span class="n">n</span><span class="si">:</span><span class="s2">5d</span><span class="si">}</span><span class="s2"> approx: </span><span class="si">{</span><span class="n">area</span><span class="si">:</span><span class="s2">.6f</span><span class="si">}</span><span class="s2"> exact: </span><span class="si">{</span><span class="mi">1</span><span class="o">/</span><span class="mi">3</span><span class="si">:</span><span class="s2">.6f</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
</code></pre></div></p>
</li>
<li>
<p>数值验证微积分基本定理。定义 <span class="arithmatex">\(F(x) = \int_0^x t^2\, dt = \frac{x^3}{3}\)</span>，并验证其导数（通过 <code>jax.grad</code> 计算）等于 <span class="arithmatex">\(x^2\)</span>。
<div class="highlight"><pre><span></span><code><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a><span class="kn">import</span><span class="w"> </span><span class="nn">jax</span>
<a id="__codelineno-1-2" name="__codelineno-1-2" href="#__codelineno-1-2"></a><span class="kn">import</span><span class="w"> </span><span class="nn">jax.numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">jnp</span>
<a id="__codelineno-1-3" name="__codelineno-1-3" href="#__codelineno-1-3"></a>
<a id="__codelineno-1-4" name="__codelineno-1-4" href="#__codelineno-1-4"></a><span class="n">F</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">x</span><span class="o">**</span><span class="mi">3</span> <span class="o">/</span> <span class="mi">3</span>
<a id="__codelineno-1-5" name="__codelineno-1-5" href="#__codelineno-1-5"></a><span class="n">dF</span> <span class="o">=</span> <span class="n">jax</span><span class="o">.</span><span class="n">grad</span><span class="p">(</span><span class="n">F</span><span class="p">)</span>
<a id="__codelineno-1-6" name="__codelineno-1-6" href="#__codelineno-1-6"></a>
<a id="__codelineno-1-7" name="__codelineno-1-7" href="#__codelineno-1-7"></a><span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="p">[</span><span class="mf">0.5</span><span class="p">,</span> <span class="mf">1.0</span><span class="p">,</span> <span class="mf">2.0</span><span class="p">,</span> <span class="mf">3.0</span><span class="p">]:</span>
<a id="__codelineno-1-8" name="__codelineno-1-8" href="#__codelineno-1-8"></a> <span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;x=</span><span class="si">{</span><span class="n">x</span><span class="si">:</span><span class="s2">.1f</span><span class="si">}</span><span class="s2"> F&#39;(x)=</span><span class="si">{</span><span class="n">dF</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="si">:</span><span class="s2">.4f</span><span class="si">}</span><span class="s2"> x^2=</span><span class="si">{</span><span class="n">x</span><span class="o">**</span><span class="mi">2</span><span class="si">:</span><span class="s2">.4f</span><span class="si">}</span><span class="s2">&quot;</span><span class="p">)</span>
</code></pre></div></p>
</li>
<li>
<p>可视化 <span class="arithmatex">\(f(x) = \sin(x)\)</span> 从 <span class="arithmatex">\(0\)</span> 到 <span class="arithmatex">\(\pi\)</span> 的曲线下面积。使用 <code>plt.fill_between</code> 填充该区域，并用黎曼和数值计算面积。
<div class="highlight"><pre><span></span><code><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a><span class="kn">import</span><span class="w"> </span><span class="nn">jax.numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">jnp</span>
<a id="__codelineno-2-2" name="__codelineno-2-2" href="#__codelineno-2-2"></a><span class="kn">import</span><span class="w"> </span><span class="nn">matplotlib.pyplot</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">plt</span>
<a id="__codelineno-2-3" name="__codelineno-2-3" href="#__codelineno-2-3"></a>
<a id="__codelineno-2-4" name="__codelineno-2-4" href="#__codelineno-2-4"></a><span class="n">x</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">jnp</span><span class="o">.</span><span class="n">pi</span><span class="p">,</span> <span class="mi">500</span><span class="p">)</span>
<a id="__codelineno-2-5" name="__codelineno-2-5" href="#__codelineno-2-5"></a><span class="n">y</span> <span class="o">=</span> <span class="n">jnp</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
<a id="__codelineno-2-6" name="__codelineno-2-6" href="#__codelineno-2-6"></a>
<a id="__codelineno-2-7" name="__codelineno-2-7" href="#__codelineno-2-7"></a><span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;purple&quot;</span><span class="p">,</span> <span class="n">linewidth</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
<a id="__codelineno-2-8" name="__codelineno-2-8" href="#__codelineno-2-8"></a><span class="n">plt</span><span class="o">.</span><span class="n">fill_between</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.2</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">&quot;purple&quot;</span><span class="p">)</span>
<a id="__codelineno-2-9" name="__codelineno-2-9" href="#__codelineno-2-9"></a><span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;Area = </span><span class="si">{</span><span class="n">jnp</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">jnp</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="p">(</span><span class="n">jnp</span><span class="o">.</span><span class="n">pi</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">500</span><span class="p">))</span><span class="si">:</span><span class="s2">.4f</span><span class="si">}</span><span class="s2"> (exact: 2.0)&quot;</span><span class="p">)</span>
<a id="__codelineno-2-10" name="__codelineno-2-10" href="#__codelineno-2-10"></a><span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</code></pre></div></p>
</li>
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