Work Done Stretching a Spring Using Hooke's Law
How to use Hooke's Law to calculate the work done in stretching a spring
Problem: Suppose a spring has natural length of 3 ft and it takes 10 ft-lb to stretch it from 5 ft to 8 ft.
How to use Hooke's Law to calculate the work done in stretching a spring
Calculating a spring constant and the work done stretching a spring
Solving for a spring constant and calculating work done stretching a spring
Review of work and average value
Finding the work to pump water out of a tank
Finding the work to pump water out of a tank
Calculating the work done moving an object with a variable force
Calculating the work done pulling a rope to the top of a building
Calculating the work done pulling a rope and weight to the top of a building
Calculating the work done pulling part of a rope to the top of a building
Review of work and average value
Using Python to calculate the amount of work done in moving a spring using Hooke's Law
Review examples covering Riemann Sums and definite integrals
Examples finding absolute maximums, antiderivatives, and definite integrals
Using integrals to solve application problems
Examples solving Definite Integrals
Solving more application problems using definite integrals
Application problems finding the average value of a function
Another example finding the average value of a function
Finding the average value of a function
Solving examples finding the area between curves
Finding the area between two curves
Finding the area between two curves
Using Reimann sums and the Fundamental Theorem of Calculus
Using Reimann sums and the Fundamental Theorem of Calculus
Evaluating a definite integral by interpreting it as an area and using geometry
Using the Fundamental Theorem of Calculus to find the derivative of a function defined using an integral
Evaluating a definite integral containing a polynomial
Evaluating a definite integral by first simplifying the integrand
Evaluating a definite integral with an exponential and sine function
Using u-substitution to evaluate an indefinite integral
Reviewing u-Substitution
Review of trigonometric substitution
Review of partial fractions
Examples finding a volume using disks, washers, and by slicing
Using the disk, washer, and shell method to find a volume of revolution
Finding volume using using disks, washers and the shell method
Using integrals to solve work problems and find the average value
Examples using integration by parts
Integrating combinations of trigonometric functions
Review of integration using trigonometric substitution
Determining divergence or convergence of improper integrals
Finding the area of a region enclosed by two curves
Finding the area of a region enclosed by two curves
Finding the area of a region enclosed by two curves
Finding the area of a region bounded by curves
Finding the volume of a non-rotational solid
Finding the volume of a non-rotational solid
Finding the volume of a solid of revolution using the disk method
Finding the volume of a solid of revolution using the disk method
Finding the volume of a solid of revolution using the washer method
Finding the volume of a solid of revolution using the washer method
Finding the volume of a solid of revolution using cylindrical shells
Finding the volume of a solid of revolution using cylindrical shells
Determining if an improper integral converges or diverges
Determining if an improper integral converges or diverges
Determining if an improper integral converges or diverges
Determining if an improper integral converges or diverges
Using the comparison test to determine if an improper integral converges or diverges
Using the comparison test to determine if an improper integral converges or diverges
Using the comparison test to determine if an improper integral converges or diverges
Using the comparison test to determine if an improper integral converges or diverges
Evaluating a definite integral by rewriting the integrand as its Maclaurin series
Finding the length of a curve given by parametric equations
Find the integral representing the surface area of a rotated parametric curve
Finding the surface area of a rotated parametric curve
Finding the area of a region defined using polar coordinates
Finding the area of a region defined using polar coordinates
Finding the area of a region defined using polar coordinates
Finding the area of a region defined using polar coordinates
Finding the area of a region defined using polar coordinates
Explaining the definition of work for constant and non-constant forces
How to calculate the work done in lifting up a rope possibly with a mass
Explaining how to use integration to calculate the work done pumping water out of a tank
Calculating the work done lifting a barbell
Calculating the work done lifting a weight
Calculating the work done pumping water out of a tank that is an upright cylinder
Calculating the work done pumping water out of a rectangular tank
Calculating the work done pumping water out of a tank shaped like a trough
Calculating the work done pumping water out of a tank shaped like a trough
Calculating the work done pumping water out of a cone-shaped tank
The definition and properties of definite integrals
Using Reimann sums and the Fundamental Theorem of Calculus
Using Reimann sums and the Fundamental Theorem of Calculus
Using u substitution to evaluate integrals and prove facts about logarithms and integrals and
Reviewing u-Substitution
Review of trigonometric substitution
Review of partial fractions
Examples finding a volume using disks, washers, and by slicing
Using the disk, washer, and shell method to find a volume of revolution
Finding volume using using disks, washers and the shell method
Using integrals to calculate work done in physics examples
Using integrals to solve work problems and find the average value
Using integrals to solve work problems and find the average value
Examples using integration by parts
Integrating combinations of trigonometric functions
Review of integration using a trigonometric substitution
Determining divergence or convergence of improper integrals
Finding the length of a three-dimensional curve
Using Green's Theorem to calculate the work done as a particle moves through a force field
Using Stokes' Theorem to express a surface integral of the curl of a vector field as a single integral
Finding the volume of a solid of revolution in Python
Using Python to calculate the work done in pumping the liquid from a tank whose ends are semicircles
Solving definite and indefinite integrals in Python
Indefinite integral examples and u-substitution
Finding antiderivatives of functions
More examples using u-Substitution
Using u-Substitution to find the Antiderivative
Solving several application problems using integrals
Examples of integration by substitution
Using the dot product to calculate work done
Using vectors to find the magnitude and direction of a resultant force
Using vectors to find the work done when a force moves an object
Calculating the work done pushing a crate up a ramp
Using a limit of Riemann sums to write an expression for the area under a graph
Evaluating a definite integral by using geometry to find the area under the curve
Evaluating a definite integral by using geometry to find the area under the curve
Using u-substitution to evaluate an indefinite integral
Evaluating an indefinite integral by simplifying a fraction to powers of x
Evaluating an indefinite integral with powers of x and using arctangent
Examples reviewing integration techniques from Calculus I
Review of partial fractions
Review of using tests to determine if a series is absolutely convergent, convergent, and divergent
Review of power series properties and writing functions as power series
Review of Taylor and Maclaurin Series and their properties
Review of using tests to determine if a series is absolutely convergent, convergent, and divergent
Examples of integration by substitution
Examples finding the area bounded by curves
Using the method of partial fractions to integrate rational functions
Using the method of partial fractions to integrate rational functions
Integrating an exponential function using u-substitution
Integrating a rational function using u-substitution
Integrating a rational function using u-substitution
Integrating a function with a square root using u-substitution
Integrating a function with sine using u-substitution
Integrating a function using u-substitution
Evaluating an integral using integration by parts
Evaluating an integral using integration by parts
Evaluating an integral using integration by parts
Evaluating an integral using integration by parts
Evaluating an integral using integration by parts
Evaluating an integral using trigonometric identities
Evaluating an integral using trigonometric identities
Evaluating an integral using trigonometric identities
Evaluating an integral using trigonometric identities
Evaluating an integral using trigonometric identities
Evaluating an integral using trigonometric substitution
Evaluating an integral using trigonometric substitution
Evaluating an integral using trigonometric substitution
Evaluating an integral using integration by parts
Evaluating an integral using integration by parts
Evaluating an integral using integration by parts
Evaluating an indefinite integral using a power series
The definition of the Riemann Sum and how it relates to a definite integral
Using Riemann sums to find the definite integral for the area between curves
Determining whether its easier to integrate along the \(x\) or \(y\)-axis to find the area between curves
Using multiple integrals to find the area between two curves
Finding the area of a region bounded by two curves
Finding the area of a region bounded by two curves
Finding the area of a region bounded by two curves
Finding an integral representing the area of a region bounded by two curves
Finding the area of a region bounded by two curves
Using the disk method to find the volume of a rotational solid
Using the disk method to find the volume of a rotational solid
Using the washer method to set up an integral for the volume of a rotational solid
Using the washer method to set up an integral for the volume of a rotational solid
Using the washer method to set up an integral for the volume of a rotational solid
Finding the volume of a non-rotation solid
Finding the volume of a non-rotational solid
Finding the volume of a non-rotational solid
Deriving the formula for the method of cylindrical shells
Explaining how to use cylindrical shells when the region is rotated around the \(y\)-axis
Using cylindrical shells to find the volume of a region rotated around the \(y\)-axis
Using cylindrical shells to find the volume of a region rotated around the \(y\)-axis
Using disks and shells to find the volume of a rotational solid
Deriving the formula for integration by parts using the product rule
Solving a definite integral with an exponential function using integration by parts
Solving a definite integral with arctangent using integration by parts
Using trigonometric identities to integrate powers of sine and cosine
Using trigonometric identities to integrate powers of secant and tangent
Using a trigonometric identity to integrate powers of sine and cosine
Using u-substitution to evaluate an definite integral with a power of a polynomial in the denominator
Using u-substitution to integrate a rational function
Using u-substitution to evaluate a definite integral with a natural logarithm
Using u-substitution on an indefinite integral with an exponential function, sine, and cosine
Using partial fraction decomposition to rewrite and then integrate a rational function
Using partial fraction decomposition to rewrite and then integrate a rational function
Using trigonometric substitution to evaluate a definite integral
Using trigonometric substitution to evaluate a definite integral
Solving an improper integral where the upper limit of integration is infinity
Solving an improper integral where the upper limit of integration is infinity
Solving an improper integral where the upper limit of integration is infinity
Solving an improper integral where the upper limit of integration is infinity
Solving an improper integral where the upper limit of integration is infinity
Solving an improper integral where the upper limit of integration is infinity
Solving an improper integral where the lower limit of integration is negative infinity
Solving a type two improper integral with a limit
Solving a type two improper integral using a limit
Solving a type two improper integral using a limit
Using the Comparison Theorem to solve an improper integral
Using the Comparison Theorem to solve an improper integral
Using the Comparison Theorem to solve an improper integral
Proving facts about antiderivatives and a physics application
Proving and then applying the Fundamental Theorem of Calculus
Examples of integration by substitution
Examples reviewing integration techniques from Calculus I
Review of the definition of a definite integral
Review of partial fractions
Review of using tests to determine if a series is absolutely convergent, convergent, and divergent
Review of power series properties and writing functions as power series
Review of Taylor and Maclaurin Series and their properties
Review of using tests to determine if a series is absolutely convergent, convergent, and divergent
Examples of integration by substitution
Examples finding the area bounded by curves
Interpreting integrals to represent areas between curves
Using the disk and washer method to calculate the volume of rotational solids
Using cylindrical shells to calculate the volume of a rotational solid
Calculating the average value of a function and proving the Mean Value Theorem for Integrals
Using the method of partial fractions to integrate rational functions
Using the method of partial fractions to integrate rational functions
Deriving and using the formula for the arc length of a curve
Using Riemann Sums and Integrals to calculate hydrostatic force
Finding the position vector function given the velocity and an initial position
Determining if a vector field is conservative and then finding a potential function
Using the Fundamental Theorem of Line Integrals to evaluate a line integral of a vector field
Using the Fundamental Theorem of Line Integrals to evaluate a line integral of a vector field
Using the Fundamental Theorem of Line Integrals to evaluate a line integral of a vector field
Finding the curl and divergence of a three dimensional vector field
Finding derivatives in Python and solving for the rate of change of a force
Using Python to approximate a definite integral using left endpoint Riemann sums
Using Python to identify the region between curves and then evaluate the area of the region
Explaining how to find the derivative and integral of matrix and vector functions
Explaining how to calculate probabilities for continuous data using integrals
Solving for a constant so that a function is a probability density function
Deriving the formula for integration by parts using the product rule
Explaining how to calculate probabilities for a uniform distribution