Area & Riemann Sums

Motivation for Integrals

• Question: What is an integral and why is it important?

  • Answer: An integral is a mathematical tool used to compute complicated expressions by breaking them down into smaller, manageable pieces.

• Examples of applications for integrals:

  • Calculating the area under a curve for a given function, e.g., y = f(x).

  • Determining the arc length of a complex curve.

  • Computing the volume of a liquid in a container with specified dimensions, or the work required to move that liquid.

  • Calculating the energy required to escape a planet's gravitational pull, such as determining Earth's escape velocity.

The Main Idea Behind Integration: Three Steps

1. Break into pieces: Divide the object or problem into n smaller parts.

2. Approximate each piece: Use a simpler, computable approximation for each small piece, leveraging the