Calculus Today

Today calculus is taught in high schools and universities across the world. With the aid of computers we can solve problems previously unsolvable, and with complicated numerical approximations, we can generate useful solutions for problems that otherwise could not be reached. But the fundamentals are still the same.

Integration

Integration is often thought of by most as calculating the area between a positive curve and the x-axis, and it is based upon a set of principles developed from the Archimedes’ work. Today we generally consider a set of rectangles, with heights determined by the function, and equal base lengths spanning the interval to be integrated (a ... b). The area under the curve lies between the lower sum (shown in red, always less than the actual area) and the upper sum (in blue, always greater), and is approximated ever more closely by these sums as the number (n) approaches infinity.

Differentiation

Differentiation is usually described as finding slopes of tangents, and is best described geometrically. Given two points on a curve, the slope of the chord that joins then is close to the slope of the curve. If we consider the effect of moving the two points closer together we see that the slope of the chord approached that of the curve.