Proposition 2. If then , and the substitution rule simply says if you let formally in the integral everywhere, what you naturally would hope to be true based on the notation actually is true. The substitution rule illustrates how the notation Leibniz invented for Calculus is incredibly brilliant.
It is said that Leibniz would often spend days just trying to find the right notation for a concept. He succeeded. As with all of Calculus, the best way to start to get your head around a new concept is to see severally clearly worked out examples.
And the best way to actually be able to use the new idea is to do lots of problems yourself! The history of the technique that is currently known as integration began with attempts to find the area underneath curves. The foundations for the discovery of the integral were first laid by Cavalieri , an Italian Mathematician, in around In order to deal with the geometrical notion of a moving point, Cavalieri worked with what he called "indivisibles". That is, if a moving point can be considered to sketch a curve, then Cavalieri viewed the curve as the sum of its points.
By this notion, each curve is made up of an infinite number of points, or "indivisibles". Likewise, the "indivisibles" that composed an area were an infinite number of lines. Though Cavalieri was not the first person to consider geometric figures in terms of the infinitesimal Kepler had done so before him , he was the first to use such a notion in the computation of areas Hooper In Figure 1. The total area of the inner rectangular regions can easily be computed by taking the sum of all the inner rectangles.
Comparing the two areas:. Using the same methodology, the ratio for a larger rectangle with a greater number of inner subdivisions is computed:. The total area of the inner regions is always one-half the area of the total rectangle. This can be shown formally by using the closed form of the summation for the numerator:.
Cavalieri now took a step of great importance to the formation of the integral calculus. He utilized his notion of "indivisibles" to imagine that there were an infinite number of shaded regions. He saw that as the individual shaded regions became small enough to simply be lines, the jagged steps would gradually define a line. As the jagged steps became a line, the shaded region would form a triangle.
As the number of shaded regions increases, the ratio remains simply one-half. He had also shown that his notion of "indivisibles" can be used to successfully describe the area underneath the curve. That is, as the areas of the rectangles turn into lines, their sum does indeed produce the area underneath the curve in this case, a line.
Cavalieri went on to use his method of "indivisibles" to find the area underneath many different curves. However, he was never able to formulate his techniques into a logically consistent foundation that others accepted. In order to do so, this technique will be applied to find the area underneath the parabola. Each rectangular region has a base of 1 unit along the x-axis and height of x 2 obtained from the definition of the parabola. The number of rectangular regions will be defined to be m.
Cavalieri again attempted to express the area underneath the curve as the ratio of an area that was already known. He considered the area enclosing all of the m rectangles.
The height of the enclosing rectangle will be m 2 , from the definition of the parabola. The ratio can now be expressed with the following equation:. Recall that the area of a rectangle is defined by the product of its base and height. The numerator is easily explained as well: each of the m rectangles has a base of 1 and a height of its x value squared. Cavalieri now proceeded to calculate the ratio for different values of m.
In doing so, he noticed a pattern and was able to establish a closed form for the ratio of the areas:. Cavalieri then utilized his important principle of "indivisibles" to make another important leap in the development of the calculus.
He noticed that as he let m grow larger, the term had less influence on the outcome of the result. In modern terms, he.
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