Lebesgue Integration and Lower Sums

calculus – how things change – continuous change

  • What is Calculus Used For? (video 8:50) – developing mathematical models, including models that are developed to avoid ethically difficult experiments. Three different examples from the field of human health are presented.
  • Learn Calculus Visually in 5 minutes (video 4:17)]] – This is a simple area of the theory behind integrating the area under the curve. Essentially, we calculate the area under curves by placing infinitely many “small” rectangles under it.

Calculus with Confidence (video 10:00) – visualization – seeing the process – getting ready to study calculus about a different perspective, a perspective that makes the subject less frightening and makes the student more successful.

Random mathematical formulæ illustrating the field of pure mathematics
  • function – equation that defines the relationship between one variable and another – for a car in a drag race – position, velocity, acceleration
  • limits
  • differentiation
  • Integration – work backwards from acceleration
  • application


differential equations

partial differential equations – Sometimes, the mathematical solutions are too complex to solve so engineers and mathematicians find ways to make the calculations less complex. The answer is “close enough” so that engineers can get on with their work.

  • Navier-Stokes equations – All of the dependent variables are functions of all four independent variables. The differential equations are therefore partial differential equations

Teen finds the ‘shape’ of our beating hearts – Kevin worked with partial differential equations. This alternative way allows him to mathematically express heartbeats and is specifically designed to include numbers that are constantly changing over time — like those numbers describing how a heart changes shape when it contracts. Instead of piling one type of mathematical expression atop another, partial differential equations allow the heart muscle and the electrical signals of the heartbeat to move together, Kevin explains.
Engineering ideas

  • Navier-Stokes equations, fluid mechanics

Learn more…