Riemann Sums and Definite Integrals: MATH 151 5.2 Problems 6-12
Using Reimann sums and the Fundamental Theorem of Calculus
Approximating Definite Integrals with Riemann Sums in Python
Author: David Manuel
In this video, it is demonstrated how to use Python to approximate a definite integral using a left endpoint Reimann sum. The presenter shows how to create a list of the left endpoint values, substitute the list of into the function, and then sum the list of the y values times the width of the rectangles.
Transcript 20
Transcript 20
Problem: Approximate the integral below using 50 left endpoint rectangles.\[\int_0^2 e^{-x^2}\, dx\]
Python Code:
from sympy import *
import numpy as np
x=symbols('x')
f=exp(-x**2)
a=0
b=2
n=50
dx=(b-a)/n
xi=np.arange(a,b,dx)
yi=[f.subs(x,i) for i in xi]
Approx=np.sum(yi)*dx
print('The area is approximately', Approx)
Open Python Notebook File
from sympy import *
import numpy as np
x=symbols('x')
f=exp(-x**2)
a=0
b=2
n=50
dx=(b-a)/n
xi=np.arange(a,b,dx)
yi=[f.subs(x,i) for i in xi]
Approx=np.sum(yi)*dx
print('The area is approximately', Approx)
Open Python Notebook File
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Mean Value Theorem and properties of a graph
Mean Value Theorem and Studying the Shape of Curves: MATH 151 4.3 Problems 1-6
Mean Value Theorem and properties of a graph
L'Hospital's Rule: MATH 151 4.4 Problem 1(a-h)
Using L'Hospital's Rule to solve limits
Integration by Substitution: MATH 171 Problems 4-6
Using u substitution to evaluate integrals and prove facts about logarithms and integrals and
Review for the Common Exam: MATH 152 Exam 1 Review Problems 11-14
Review of work and average value
Review for the Common Exam: MATH 152 Exam 1 a Review Problems 1-4
Reviewing u-Substitution
Review for the Common Exam: MATH 152 Exam 1 b Review Problems 29-33
Review of trigonometric substitution
Review for the Common Exam: MATH 152 Exam 1 b Review Problems 34 & 35
Review of partial fractions
Review for the Common Exam: MATH 152 Exam 3 b Review Problems 4 & 5
Review of using tests to determine if a series is absolutely convergent, convergent, and divergent
Review for the Common Exam: MATH 152 Exam 2 b Review Problems 8 & 9
Review of improper integrals and the Comparison Theorem
Review for the Common Exam: MATH 152 Exam 3 c Review Problems 10-13
Review of power series properties and writing functions as power series
Review for the Common Exam: MATH 152 Exam 3 c Review Problems 19 & 20
Review of Taylor Polynomials and Taylor's Inequality
Review for the Common Exam: MATH 152 Exam 3 c Review Problems 4 & 5
Review of using tests to determine if a series is absolutely convergent, convergent, and divergent
Taylor and Maclaurin Series: MATH 152 11.10 Problems 4-10
Finding Taylor and Maclaurin Series for functions
Integral and Comparison Tests: MATH 172 Problems 4-7
Proving the Integral, Comparison, and Limit Comparison Tests
Area Between Curves Using Multiple Integrals
Using multiple integrals to find the area between two curves
Cylindrical Shells when Revolving Around y-axis
Explaining how to use cylindrical shells when the region is rotated around the \(y\)-axis
Work Done Stretching a Spring Using Hooke's Law
How to use Hooke's Law to calculate the work done in stretching a spring
Work Done Pumping Water Out of a Tank
Explaining how to use integration to calculate the work done pumping water out of a tank
Work and Average Value: MATH 152 Problems 1-9
Using integrals to solve work problems and find the average value
Trapezoid Rule: MATH 172 Problems 1 & 2
Covering the Trapezoid Rule for approximating the value of an integral
WIR2 20B M251 V7
Finding the limit of a three-dimensional vector function
WIR2 20B M251 V10
Finding the length of a three-dimensional curve
WIR3 20B M251 V9
Finding the partial derivatives of a function of two variables
WIR3 20B M251 V11
Finding all second-order partial derivatives of a function of two variables
WIR5 20B M251 V7
Evaluating a double integral over a given region
WIR5 20B M251 V10
Evaluating a double integral by reversing the order of integration
WIR8 20B M251 V4
Evaluating a line integral along a curve
WIR8 20B M251 V5
Evaluating a line integral along a curve in space
WIR9 20B M251 V5
Finding the curl and divergence of a three dimensional vector field
WIR10 20B M251 V4
Evaluating a surface integral of a vector field
WIR10 20B M251 V7
Using Stokes' Theorem to express a surface integral of the curl of a vector field as a single integral
Scalar-Valued IVP
Showing how the exponential function is the solution to an initial value problem
Continuous Probabilities
Explaining how to calculate probabilities for continuous data using integrals
Probability Density Function Exercise 2
Solving for a constant so that a function is a probability density function
Uniform Probability
Explaining how to calculate probabilities for a uniform distribution
Math 140 5.4 Exercise 3
Rewriting a function with a fractional exponent in radical form
WIR4 20B M150 V21
Solving an equation with logarithmic functions
WIR4 20B M150 V22
Solving an equation with logarithmic functions
WIR4 20B M150 V23
Solving an equation with logarithmic functions
WIR4 20B M150 V24
Solving an equation with logarithmic functions
WIR8 20B M150 V11
Solving an equation with logarithmic functions
Math 140 Exam 3 Review Exercise 13
Stating the domain in interval notation of a function with radicals, fractions, and exponentials
Math 140 Exam 3 Review Exercise 17
Calculating continuously compounded interest for money in an account
Math 140 Exam 3 Review Exercise 19
Writing logarithmic equations in exponential form
Math 140 Exam 3 Review Exercise 20
Solving equations with exponentials and logarithms
Math 140 Exam 3 Review Exercise 21
Solving equations with exponentials and logarithms
Math 140 Exam 3 Review Exercise 23
Solving equations with exponentials and logarithms
Composition of Logarithmic and Exponential Functions
Finding the composition of two functions with exponentials and logarithms
An Equation with an Exponential Function
Solving a equation with an exponential function
An Equation with Exponential Functions of Different Bases
Solving an equation with two exponential functions with different bases
Factoring a Quadratic-Like Equation with Exponentials
Solving an equation with exponential functions by factoring
Factoring an Equation with Exponentials
Solving a equation with exponential functions by factoring
Population Model with Exponential Growth
Finding the size of a population of rabbits with exponential growth
Population Model with Exponential Decay
Finding the size of a population of rabbits with exponential decay
Horizontal Asymptote of an Exponential Function
Finding the horizontal asymptote of an exponential function
Horizontal Asymptote of an Exponential Function (2)
Finding the horizontal asymptote of an exponential function
Horizontal Asymptote of an Exponential Function (3)
Finding the horizontal asymptote of an exponential function
WIR 20B M152 V53
Determining if a series converges or diverges and find the sum if it converges
WIR 20B M152 V58
Using the comparison test to determine if a series converges or diverges
WIR 20B M152 V59
Using the limit comparison test to determine if a series converges or diverges
Introduction and Installation of Python
An introduction to installing and getting started with Python
Creating and Manipulating Symbolic Expressions in Python
Creating symbolic expression in Python and then factoring, expanding, and simplifying the expressions
Plotting Expressions and Solving Equations Numerically in Python
Plotting an expression in Python and finding a numerical approximation for the solution
Plotting Multiple Expressions in One Graph in Python
Graphing a piecewise function using Python
Solving Equations Symbolically in Python
Solving equations symbolically in Python and interpreting the results
Using Variables in Python
Defining variables in Python and using the variables in equations
Simplifying Difference Quotients in Python
Using Python to simplify a difference quotient and find a derivative using the limit definition
Tangents to Parametrized Curves at a Given Point in Python
Using Python to plot a parametrized curve and its tangent line at a point
Parametric Plots and Tangents at a Given t-value in Python
Using Python to find the tangent line to a parametric equation and plot the two graphs
List Comprehension and Numerical, Graphical, and Symbolic Limits in Python
Using Python to numerically estimate a limit, graphically estimate a limit, and find the exact limit
List Comprehension and Patterns in Higher Derivatives in Python
Using Python's list comprehension tool to find several higher order derivatives of a function at once
Implicit Plots and Implicit Differentiation in Python
Using Python to plot an implicit curve and find a tangent line using implicit differentiation
Finding Absolute (Global) Extrema in Python
Using Python to find the absolute maximum and minimum of a function on a closed interval
Finding and Graphing Tangent Lines to a Curve in Python
Using Python to find the equation of the tangent line to a curve and graphing the result
Finding Derivatives in Python
Finding derivatives in Python and solving for the rate of change of a force
Finding Horizontal Tangent Lines in Python
Finding when the tangent line to a function is horizontal in Python
Finding Intervals of Increase/Decrease and Concavity in Python
Using Python to find the intervals where a function is increasing or decreasing
Word Problems and the Algorithmic Process for Multistep Problem Solving in Python
Solving a multistep word problem in Python and graphing the resulting function
For Loops and Recursive Sequences in Python
Using for loops in Python to generate a recursively defined sequence
Generating and Plotting Partials Sums of a Series in Python
Using Python to calculate the partial sums of a series to estimate the value of the series
Generating and Plotting Sequences in Python
Using Python to numerically, graphically, and analytically find the limit of a sequence
Plotting a Function and Partial Sums of its Power Series in Python
Video 29 in a series of Python instructional videos
Radius and Interval of Convergence in Python
Using Python to find the radius and interval of convergence of a power series with the Ratio Test
Domain of Combinations of Functions Video 4
Finding the domain of a function with a root and a logarithm in the denominator
Properties of Radical Functions and Expressions Video 2
How to rewrite radical terms as power terms
Properties of Power and Exponential Expressions Video 1
Properties of exponents applied to power and exponential expressions
Properties of Power and Exponential Expressions Video 2
Properties of multiplying and simplifying power and exponential expressions
Properties of Power and Exponential Expressions Video 3
Properties of exponents applied to power and exponential expressions
Properties of Power and Exponential Expressions Video 6
Simplifying power and exponential expressions using properties of exponents
Properties of Power and Exponential Expressions Video 7
Simplifying power and exponential expressions using properties of exponents
Properties of Power and Exponential Expressions Video 9
Simplifying an expression with both power and exponential functions using properties of exponents
Properties of Power and Exponential Expressions Video 4
Properties of exponential growth and decay functions
Properties of Power and Exponential Expressions Video 5
Properties of the exponential function with base e
Solving Power and Exponential Equations Video 1
Explaining how to solve equations with power and exponential terms
Solving Power and Exponential Equations Video 2
Solving equations with exponential terms
Solving Power and Exponential Equations Video 3
Solving equations with base e exponential terms
Properties of Logarithmic Functions and Expressions Video 1
Explaining the terminology and rules of logarithms
Solving Logarithmic Equations Video 1
Explaining how to solve logarithmic equations
Solving Logarithmic Equations Video 2
Solving logarithmic equations with a single logarithm term
Solving Logarithmic Equations Video 3
Solving logarithmic equations with multiple logarithm terms
Solving Logarithmic Equations Video 4
Solving logarithmic equations with multiple logarithm terms on both sides
Properties of Power and Exponential Expressions Video 1
Properties of exponents applied to power and exponential expressions
Properties of Power and Exponential Expressions Video 2
Properties of multiplying and simplifying power and exponential expressions
Properties of Power and Exponential Expressions Video 3
Properties of exponents applied to power and exponential expressions
Properties of Power and Exponential Expressions Video 4
Properties of exponential growth and decay functions
Properties of Power and Exponential Expressions Video 5
Properties of the exponential function with base e
Properties of Logarithmic Functions and Expressions Video 1
Explaining the terminology and rules of logarithms
Derivatives of Exponential Functions
Explaining the formulas for the derivatives of exponential functions
Probability Density Function Exercise 2
Solving for a constant so that a function is a probability density function
Review for the Common Exam: MATH 151 Exam 3 Review Problems 8-15
Review of limits and derivatives of inverse trigonometric functions