{ "cells": [ { "cell_type": "markdown", "id": "540516aa", "metadata": {}, "source": [ "# Integrals" ] }, { "cell_type": "markdown", "id": "2e09318a", "metadata": {}, "source": [ "**Instructor:** Sophie Wynn, Climate Science PhD Student (srwynn@ucsd.edu)\n", "\n", "**TAs:** Luisa Watkins (l1watkins@ucsd.edu) and Lauren Harvey (lrharvey@ucsd.edu)\n", "\n", "This lecture introduces integration and covers Riemann Sums, the FTC, and integration by parts.\n", "\n", "Lecture Notes inspired by Luisa Watkins and 3Blue1Brown." ] }, { "cell_type": "markdown", "id": "1474c961", "metadata": {}, "source": [ "## 1 | Integration \n", "\n", "The **integral** measures the total accumulation of a quantity over time, space etc. For example an integral can compute area, distance, volume, mass, work etc!\n", "\n", "**To get an understanding of what this means, let's go through an example:**\n" ] }, { "cell_type": "markdown", "id": "adbcf335", "metadata": {}, "source": [ "### 1.1 Traveling in a Car \n", "\n", "Suppose our velocity is given by:\n", "\n", "$$\n", "v(t) = t(8 - t), \\quad 0 \\leq t \\leq 8\n", "$$\n", "\n", "But we were only keeping track of the velocity every second. \n", "\n", "Sampled values of velocity\n", "\n", "| Time \\(t\\) (s) | Velocity \\(v(t)\\) (m/s) |\n", "|----------------|--------------------------|\n", "| 0 | 0 |\n", "| 1 | 7 |\n", "| .. | ... |\n", "| 4 | 16 |\n", "| 8 | 0 |\n", "\n", "Question: **How far did the car travel between \\(t = 0\\) and \\(t = 8\\)?**\n", "\n", "\n", "\n", "We’ll compute the **distance traveled** in two ways: \n", "1. Using a **table of sampled values** and **Riemann sums**. \n", "2. Using the **exact integral**. \n" ] }, { "cell_type": "code", "execution_count": 3, "id": "6aaaf505", "metadata": {}, "outputs": [ { "data": { "image/png": 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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "import numpy as np\n", "import matplotlib.pyplot as plt\n", "\n", "def u(t):\n", " return t * (8 - t)\n", "\n", "time_samples = np.array([0, 1, 4, 8])\n", "u_samples = u(time_samples)\n", "\n", "t_cont = np.linspace(0, 8, 400)\n", "u_cont = u(t_cont)\n", "\n", "# Left-Riemann sum setup \n", "dt = 1\n", "t_left = np.arange(0, 8, dt) # left endpoints: 0,1,2,...,7\n", "u_left = u(t_left)\n", "\n", "# Compute left Riemann sum\n", "areas = u_left * dt\n", "total_left_riemann = np.sum(areas)\n", "\n", "plt.figure(figsize=(12,5))\n", "\n", "\n", "# plot continuous velocity curve\n", "plt.plot(t_cont, u_cont, \"b-\", linewidth=2, label=r\"$u(t) = t(8-t)$\")\n", "\n", "# plot left endpoints\n", "plt.scatter(t_left, u_left, color=\"red\", zorder=5, label=\"Left endpoints\")\n", "\n", "# draw rectangles\n", "for i in range(len(t_left)):\n", " plt.bar(t_left[i], u_left[i], width=dt, align=\"edge\",\n", " alpha=0.35, edgecolor=\"k\", color=\"skyblue\")\n", "\n", "plt.title(f\"Left-Riemann Sum (Δt=1)\")\n", "plt.xlabel(\"Time (s)\")\n", "plt.ylabel(\"Velocity $v(t)$ (m/s)\")\n", "plt.grid(True)\n", "plt.legend()\n", "plt.tight_layout()\n", "plt.show()" ] }, { "cell_type": "markdown", "id": "80f98438", "metadata": {}, "source": [ "### 1.2 Riemann Sum\n", "\n", "\n", "To compute distance given the velocity graph aove, we can first use **Riemann sums**. \n", "We approximate the total distance by computing the **area of rectangles** under the curve. \n", "\n", "- The area of each rectangle is **base × height**. \n", "- The base is $(\\Delta t = 1)$ ie (s). \n", "- The height is the velocity ie (m/s) at the **left endpoint** of each interval. \n", "\n", "**Thus the area will give us meters**\n", "\n", "So, we can sum up the areas of all rectangles to get an approximation of the total distance traveled.\n", "\n", "\n", "\n", "If we divide the interval into equal parts of width $(\\Delta t)$, the **Riemann sum** is:\n", "\n", "$$\n", "S = \\sum_{i=0}^{n-1} v(t_i)\\Delta t\n", "$$\n", "\n", "where $(t_i)$ is the left endpoint of each interval.\n", "\n", "\n", "\n", "### 1.3 Car example (with Δt = 1)\n", "\n", "\n", "Here every change in time is $\\Delta t = 1$. \n", "So the Riemann sum is:\n", "\n", "$$\n", "S = (v(0)\\cdot 1 + v(1)\\cdot 1 + v(2)\\cdot 1 + v(3)\\cdot 1 + v(4)\\cdot 1 + v(5)\\cdot 1 + v(6)\\cdot 1 + v(7)\\cdot 1)\n", "$$\n", "\n", "Substitute $v(t) = t(8-t)$:\n", "\n", "$$\n", "S = (0 + 7 + 12 + 15 + 16 + 15 + 12 + 7) = 84\n", "$$\n", "\n", "So, the left Riemann sum approximation of the total distance is:\n", "\n", "$$\n", "\\boxed{84 \\text{ meters}}\n", "$$" ] }, { "cell_type": "markdown", "id": "dc878d18", "metadata": {}, "source": [ "**Imagine if, instead of using a finite** $\\Delta t = 1$ **second, we could measure the velocity at** ***every instant in time***. **Then our rectangles would get smaller and smaller, and the sum of their areas would converge to the exact distance traveled!! This is exactly what** ***integration*** **allows us to do.**\n", "\n", "---\n", "\n", "## 2 | Introducing the Integral\n", "\n", "\n", "If we let the intervals become **infinitely small** ($\\Delta t \\to 0$), the Riemann sum **converges** to the exact total accumulation, which is written as an **integral**:\n", "\n", "$$\n", "\\text{Accumulated quantity} = \\int_a^b f(t) \\, dt\n", "$$\n", "\n", "Where: \n", "\n", "- $\\int$ is the **integration operator**, \n", "- $a$ and $b$ are the **lower and upper bounds** of the interval, \n", "- $f(t)$ is the function being accumulated, \n", "- $dt$ represents an **infinitesimally small change** in the independent variable.\n", "\n", "\n", "\n", "### 2.1 Applying to our car example\n", "\n", "For the car’s velocity $v(t)$ from $t = 0$ to $t = 8$ s:\n", "\n", "$$\n", "v(t) = t(8-t)\n", "$$\n", "\n", "The **distance traveled** is:\n", "\n", "$$\n", "\\text{Distance} = \\int_0^8 v(t) \\, dt\n", "$$\n", "\n", "Using the **exact integral**, we get:\n", "\n", "$$\n", "\\int_0^8 t(8-t) \\, dt = 85.33 \\text{ m}\n", "$$\n", "\n", "\n", "So, the true of the total distance is:\n", "\n", "$$\n", "\\boxed{85.33 \\text{ meters}}\n", "$$\n", "\n", "\n", "This shows how letting $\\Delta t \\to 0$ turns the **approximation into the true value**.\n", "\n", "\n", "---\n" ] }, { "cell_type": "markdown", "id": "a1abb316", "metadata": {}, "source": [ "### 2.2 Computing the Integral\n", "\n", "$$\n", "\\int_0^8 t(8-t) \\,dt\n", "$$\n", "\n", "\n", "To compute an integral we **take the antiderivative** of the function and then compute the bounds (if stated). \n", "\n", "\n", "#### 2.2.1 Step 1: Expand the function\n", "\n", "\n", "$$\n", "t(8-t) = 8t - t^2\n", "$$\n", "\n", "\n", "#### 2.2.2 Step 2: Apply the Power Rule for Integration\n", "\n", "\n", "Recall the **power rule**, from derivatives, the same rule applies integration but in reverse:\n", "\n", "$$\n", "\\int t^n \\, dt = \\frac{t^{n+1}}{n+1} + C, \\quad n \\neq -1\n", "$$\n", "\n", "Apply it to each term:\n", "\n", "$$\n", "\\int (8t - t^2) \\, dt = 4t^2 - \\frac{t^3}{3} + C\n", "$$\n", "\n", "This is the **antiderivative** $F(t)$.\n", "\n", "\n", "#### 2.2.3 Step 3: Evaluate at the Bounds (Fundamental Theorem of Calculus)\n", "\n", "\n", "\n", "The **Fundamental Theorem of Calculus** says:\n", "\n", "$$\n", "\\int_a^b f(t)\\, dt = F(b) - F(a)\n", "$$\n", "\n", "\n", "$$\n", "F(t) = 4 t^2 - \\frac{t^3}{3}, \\quad a=0, b=8\n", "$$\n", " \n", "Compute:\n", "\n", "$$\n", "\\int_0^8 (8t - t^2) \\, dt = F(8) - F(0) = \\left(256 - \\frac{512}{3}\\right) - 0 = \\frac{256}{3} = 85.33 \n", "$$\n" ] }, { "cell_type": "markdown", "id": "02d65746", "metadata": {}, "source": [ "### 2.3 The Fundamental Theorem of Calculus\n", "\n", "The FCT is key to understanding derivatives and antiderivatives! \n", "\n", "- In our example, $F(t) = 4t^2 - \\frac{t^3}{3}$ also represents the **path function**, commonly written as $x(t)$ in physics or calculus. \n", "- Recall from derivatives: \n", "\n", "$$\n", "\\frac{dx}{dt} = v(t)\n", "$$\n", "\n", "which says the derivative of the path gives the **velocity**. \n", "\n", "- Here, we went in the opposite direction: **starting from velocity** $v(t)$, we computed the **integral** to recover the **path** $x(t)$. \n", "\n", "In other words:\n", "\n", "- **Derivative:** path $\\to$ velocity \n", "- **Integral:** velocity $\\to$ path \n", "\n", "This shows how **integration accumulates small changes in velocity** over time to give the **total displacement**, just like summing the tiny rectangles in a Riemann sum. \n", "\n", "Thus, **derivatives and integrals are inverse operations**, and the Fundamental Theorem of Calculus makes this connection precise. \n", "\n", "- Derivative: $v(t) = \\frac{dx}{dt}$ \n", "- Integral: $x(b) - x(a) = \\int_a^b v(t) \\, dt$\n" ] }, { "cell_type": "markdown", "id": "e67a40c0", "metadata": {}, "source": [ "---\n", "## 3 | Common Integration Formulas\n", "\n", "\n", "Here is a list of commonly used integrals and rules for reference:\n", "\n", "---\n", "\n", "### 3.1 Power Rule\n", "\n", "For $n \\neq -1$:\n", "\n", "$$\n", "\\int x^n \\, dx = \\frac{x^{n+1}}{n+1} + C\n", "$$\n", "\n", "---\n", "\n", "### 3.2 Exponential Functions\n", "\n", "\n", "$$\n", "\\int e^x \\, dx = e^x + C\n", "$$\n", "\n", "$$\n", "\\int a^x \\, dx = \\frac{a^x}{\\ln(a)} + C, \\quad a > 0, a \\neq 1\n", "$$\n", "\n", "---\n", "\n", "### 3.3 Logarithmic Function\n", "\n", "\n", "$$\n", "\\int \\frac{1}{x} \\, dx = \\ln|x| + C\n", "$$\n", "\n", "---\n", "\n", "### 3.4 Sine and Cosine\n", "\n", "$$\n", "\\int \\sin(x) \\, dx = -\\cos(x) + C\n", "$$\n", "\n", "$$\n", "\\int \\cos(x) \\, dx = \\sin(x) + C\n", "$$\n", "\n", "---\n", "\n", "### 3.5 Other Trigonometric Functions\n", "\n", "\n", "$$\n", "\\int \\sec^2(x) \\, dx = \\tan(x) + C\n", "$$\n", "\n", "$$\n", "\\int \\csc^2(x) \\, dx = -\\cot(x) + C\n", "$$\n", "\n", "$$\n", "\\int \\sec(x)\\tan(x) \\, dx = \\sec(x) + C\n", "$$\n", "\n", "$$\n", "\\int \\csc(x)\\cot(x) \\, dx = -\\csc(x) + C\n", "$$\n", "\n", "---\n", "\n", "### 3.6 Linear Combination Rule\n", "\n", "If $f(x)$ and $g(x)$ are integrable and $a,b$ are constants:\n", "\n", "$$\n", "\\int \\big(a f(x) + b g(x)\\big) \\, dx = a \\int f(x)\\,dx + b \\int g(x)\\,dx\n", "$$\n", "\n", "---\n", "\n", "### 3.7 Integration by Substitution (u-substitution)\n", "\n", "Used for composite functions:\n", "\n", "$$\n", "\\int f(g(x)) g'(x)\\, dx = \\int f(u)\\, du, \\quad u = g(x)\n", "$$\n", "\n", "Example:\n", "\n", "$$\n", "\\int \\sin(3x) \\, dx = -\\frac{1}{3} \\cos(3x) + C\n", "$$\n", "\n", "\n", "---\n", "\n", "### 3.8 Inverse Trigonometric Functions\n", "\n", "$$\n", "\\int \\frac{1}{\\sqrt{1-x^2}} \\, dx = \\arcsin(x) + C\n", "$$\n", "\n", "$$\n", "\\int \\frac{1}{1+x^2} \\, dx = \\arctan(x) + C\n", "$$\n", "\n", "$$\n", "\\int \\frac{1}{x\\sqrt{x^2-1}} \\, dx = \\operatorname{arcsec}(|x|) + C\n", "$$\n", "\n", "---\n", "\n", "### 3.9 Hyperbolic Functions\n", "\n", "$$\n", "\\int \\sinh(x) \\, dx = \\cosh(x) + C\n", "$$\n", "\n", "$$\n", "\\int \\cosh(x) \\, dx = \\sinh(x) + C\n", "$$\n", "\n", "---\n", "\n", "### 3.10 Special Rules / Tricks\n", "\n", "- **Constant Multiple Rule**: $\\int a f(x)\\, dx = a \\int f(x)\\, dx$ \n", "- **Sum/Difference Rule**: $\\int (f(x) \\pm g(x)) \\, dx = \\int f(x) dx \\pm \\int g(x) dx$ \n", "\n", "\n", "---" ] }, { "cell_type": "markdown", "id": "90e344a7", "metadata": {}, "source": [ "### 3.11 Integration by Parts\n", "\n", "For functions made up of two factors, the **integration by parts formula** states that for functions $u(x)$ and $v(x)$ that are differentiable:\n", "\n", "$$\n", "\\int u(x) v'(x) \\, dx = u(x)v(x) - \\int u'(x) v(x) \\, dx\n", "$$\n", "\n", "- $u(x)$: function we differentiate \n", "- $v'(x)$: function we integrate \n", "\n", "\n", "### 3.11.1 Example\n", "\n", "Suppose we want to compute:\n", "\n", "$$\n", "\\int x \\sin(kx) \\, dx\n", "$$\n", "\n", "1. **Choose parts**:\n", "\n", "- Let $u(x) = x \\quad \\Rightarrow \\quad u'(x) = 1$ \n", "- Let $v'(x) = \\sin(kx) \\quad \\Rightarrow \\quad v(x) = -\\frac{1}{k} \\cos(kx)$\n", "\n", "2. **Apply the formula**:\n", "\n", "$$\n", "\\int x \\sin(kx) \\, dx = u(x)v(x) - \\int u'(x) v(x) \\, dx\n", "$$\n", "\n", "Substitute:\n", "\n", "$$\n", "\\left( x \\cdot -\\frac{1}{k} \\cos(kx) \\right) - \\int 1 \\cdot \\left( -\\frac{1}{k} \\cos(kx) \\right) dx\n", "$$\n", "\n", "Simplify:\n", "\n", "$$\n", "-\\frac{x}{k} \\cos(kx) + \\frac{1}{k} \\int \\cos(kx) \\, dx\n", "$$\n", "\n", "3. **Integrate remaining term**:\n", "\n", "$$\n", "\\int \\cos(kx) \\, dx = \\frac{1}{k} \\sin(kx)\n", "$$\n", "\n", "So the final result:\n", "\n", "$$\n", "-\\frac{x}{k} \\cos(kx) + \\frac{1}{k^2} \\sin(kx) + C\n", "$$" ] }, { "cell_type": "markdown", "id": "e1e9e324", "metadata": {}, "source": [ "## Practice Problems\n", "\n", "**Question 1:** \n", "\n", "Express velocity $v$ , at time $t_1$ , in terms of $v$ at $t_0$ and acceleration $a(t)$\n", "\n", "**Question 2:** \n", "\n", "Find the **mass** $(m)$ of the container described belowe in terms of density ($\\rho$) using the fact that $m =\\int \\rho dV$\n", "\n", "\n", "The container is:\n", "\n", "- 1 m long \n", "- 3 m tall \n", "- Width changes with height \\(z\\) as: $w(z) = (z + 1)^2$ \n", "- The container has a **constant density** ($\\rho$)\n", "\n", "\n", "\n", "**Question 3:** \n", "\n", "Calculate:\n", "\n", "$$\n", "\\int x e^x \\, dx\n", "$$\n", "\n" ] } ], "metadata": { "kernelspec": { "display_name": "workshop-docs", "language": "python", "name": "python3" }, "language_info": { "name": "python", "version": "3.11.13" } }, "nbformat": 4, "nbformat_minor": 5 }