The origins of calculus go back at least 2500 years to the ancient Greeks, who found areas using the "method of exhaustion." They developed techniques to find the area of curved figures, which later evolved into what we now know as integral calculus.
The Greeks knew how to find the area of any polygon by dividing it into triangles and adding the areas of these triangles.
For a triangle with base \(b\) and height \(h\), the area is: \[A = \frac{1}{2} \times b \times h\]
The Greek method of exhaustion was to inscribe polygons in the figure and let the number of sides of the polygons increase. As the number of sides increases, the area of the polygon approaches the area of the circle.
The area of a circle with radius \(r\) is: \[A = \pi r^2\]
This can be found as the limit: \[A = \lim_{n \to \infty} A_n\]
where \(A_n\) is the area of the inscribed regular polygon with \(n\) sides.
Area of inscribed polygon: 0
Area of circle: 0
Difference: 0
To find the area under a curve \(y = f(x)\) from \(x = a\) to \(x = b\), we can approximate the area using rectangles and then take the limit as the number of rectangles increases.
The area under the curve is given by the definite integral: \[A = \int_{a}^{b} f(x) \, dx\]
This can be approximated by a sum of rectangles: \[A \approx \sum_{i=1}^{n} f(x_i) \Delta x\]
where \(\Delta x = \frac{b-a}{n}\) and \(x_i\) is a point in the \(i\)-th subinterval.
Approximate area: 0
Actual area: 0
Error: 0
The techniques developed for finding areas will also enable us to compute: