Grant introduces the Essence of Calculus series and outlines its goal to demystify calculus concepts.

• The series will consist of 10 videos, released daily, to help viewers understand the essence of calculus.
• The focus is on explaining core calculus ideas in a visual way rather than memorizing formulas.
• Viewers should feel empowered to feel as though they could have invented calculus themselves.

Grant uses the problem of finding a circle's area to introduce calculus concepts like integrals and derivatives.

• The area of a circle is typically known as pi times the radius squared, but the video explores the reasoning behind this formula.
• Grant suggests slicing a circle into concentric rings to respect its symmetry, an approach that is favored in mathematical problem-solving.
• By straightening out these rings and approximating them as rectangles, we can begin to understand the circle's area.

The approximation of a circle's area is explained using concentric rings and the concept of an integral.

• Each ring's area is approximated by its circumference multiplied by a small thickness, 'dr', which is a standard calculus notation.
• The sum of the areas of these rings is visualized as rectangles under the graph of the function 2 pi r.
• The aggregate area of the rectangles approaches the triangle's area, which correlates with the circle's area formula, pi r squared.

The transition from an approximate to a precise understanding of a circle's area is a core aspect of calculus.

• As 'dr' gets smaller, the approximation for the circle's area becomes more accurate.
• The sum of the areas of the rectangles for smaller 'dr' values represents the area under the function's graph.
• This transition is subtle and highlights how calculus allows us to move from approximation to precision.

Introducing the concept of an integral function through the area under a graph.

• The integral function A(x) represents the area under a graph from a fixed point to a variable endpoint.
• The integral of x squared is used as an example to illustrate the challenge of finding the area under a parabola.
• The integral function is hard to find directly, but exploring its relationship with the graph's function offers clues to its nature.

Derivatives are introduced as a way to understand the relationship between tiny changes in a function.

• The derivative dA/dx represents the ratio of a tiny change in area to the tiny change in x that caused it.
• This ratio is approximately equal to the height of the graph at the point x, a concept that becomes more accurate with smaller values of dx.
• Understanding derivatives is essential for solving integral problems and finding areas under curves.

The Fundamental Theorem of Calculus is introduced, connecting integrals and derivatives.

• The theorem demonstrates how derivatives and integrals are inverses of each other.
• It provides a way to solve for an unknown integral function by understanding its derivative.
• This relationship is key to solving many problems in calculus and is central to the subject.

Grant concludes the first video and thanks Patreon supporters for their contribution.

• The series aims to make viewers feel capable of discovering calculus through right visualizations and playful exploration.
• The details of calculus will unfold throughout the series, focusing on derivatives and integrals.
• Grant expresses gratitude to Patreon supporters for their financial support and suggestions during the series' development.