{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# More with the Definite Integral\n", "\n", "$$\\int_a^b f(x) dx = \\lim_{n \\to \\infty}\\sum_{i = 1}^n f(x_i^*) \\Delta x$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**WARM UP**\n", "\n", "1. Evaluate the following definite integral. Interpret the integral as an area, and describe the region with a graph.\n", "\n", "$$\\int_{0} ^4 8x^3 - x^2 dx$$\n", "\n", "2. Suppose the function $8x^3 - x^2$ represents the velocity of a car in miles per hour at a time $x$. Interpret the integral: $\\int_{0} ^4 8x^3 - x^2 dx$\n", "\n", "3. Suppose the function $8x^3 - x^2$ represents the rate at which water flows through a canal in ft$^3$. Interpret the integral: $\\int_{0} ^4 8x^3 - x^2 dx$." ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "%matplotlib inline\n", "import matplotlib.pyplot as plt\n", "import numpy as np\n", "import pandas as pd" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### More Rules of Integrals\n", "\n", "
| $\\displaystyle \\int x^n dx = \\frac{x^{n+1}}{n+1} + C$ | \n", "$\\displaystyle \\int e^{ax} dx = \\frac{1}{a} e^{ax} + C$ | \n", "$\\displaystyle \\int \\ln(x)dx = x\\ln(x) - x + C$ | \n", "
| $\\displaystyle \\int \\sin(x) dx = -\\cos(x) + C$ | \n", "$\\displaystyle \\int \\cos(x) dx = \\sin(x) + C$ | \n", "$\\displaystyle \\int a^x dx = \\frac{a^x}{\\ln a} + C$ | \n", "