How Much Area is Hiding Under that Curve?
Last week’s introduction to differential calculus explained how to find the slope of the tangent at any point along a simple curved function.
This week we begin working at finding the area underneath a curve. “Why bother?” is the first question that people usually ask. “It’s the important concept to introduce integral calculus” is not a very satisfying answer.
"Image of Parabola with Area Under the Curve" Image by Mike DeHaan
The Importance of the Area Under a Curve
Integral calculus is all about knowing the rate at which something happens, and then calculating the total amount of “something” that has happened.
A simple example comes when travelling. If you can bicycle at a steady 20Km per hour, how far will you get in 3 hours? The trivially simple answer is “60Km” = 20Km/hour * 3 hours.
A similar question is, “How many products can the factory make in a day if it makes 2.7 units per minute”? These are simple because the rate is a constant per unit of time.
The problem is more complex if the rate changes. The classic science question is “How far does an object fall in a given amount of time in Earth’s gravity, assuming no wind resistance”? The acceleration due to gravity is 9.8m/sec^2 (9.8 metres per second squared). The speed is constantly increasing, so we cannot simply multiply the rate (speed) times time to get the total distance.
Geometry can Approximate the Area Under a Curve
"Image of Parabola Showing Approximate Area Under the Curve" Image by Mike DeHaan
This image shows the function y = 3*x^2 + 2*x + 1. In this image, the approximate area under the parabola’s curve is shown by the four green rectangles and five yellow triangular areas.
The six points shown are at (0, 1), although it looks like (0, 0), (1, 6), (2, 17), (3, 34), (4, 57) and (5, 86).
The first green rectangle’s coordinates are (1, 0), (1, 6), (2, 6) and (2, 0). Its area is (6-0)*(2-1) = 6. Each of the other three rectangles has a base length of ’1′; their heights are 17, 34 and 57. The total area of the rectangles is 6+17+34+57 = 114.
Although the graph shows straight line segments for the parabola, the mathematical function truly is a smooth continuous curve. So the yellow triangular areas simply approximate the actual curve. The brown curve is a rough guess at the actual smooth parabolic function.
The area of a right-angle triangle is “one-half of the base times the height”. Each of these triangles is indeed right-angled. The points for the largest triangle, at the right end of the graph, comes from the points (4, 57), (5, 86) and (5, 57). The height is (86-57)=29; the base is 1; so the area is 29/2 = 14.5. The total area of all the triangles is (5 + 11 + 17 + 23 + 29)/2 = 42.5.
Therefore an approximation of the complete area under the curve is the area of the rectangles plus triangles: 114 + 42.5 = 156.5.
Let’s say that a “slice” of area is the triangle plus rectangle, if any, above the base shown by the change in the ‘x’ value.
Accuracy, or a Lack of It
We could have made a first approximation by thinking of the whole area as one triangle, with the points (0, 1), (5, 86) and (5, 1). The base would be 5, the height 85, and the area 5*85/2 = 212.5. The faint purple line shows how much extra area is included by this approximation.
The error is caused by the difference between the function’s curve versus the straight line drawn between two neighbouring points in the graph.
We can make better approximations by calculating more points along the parabola. Readers should feel free to build a spreadsheet and populate it with ‘x’ changing by 0.5, 0.1, or 0.001. Each time the change in ‘x’ becomes smaller, the straight-line “interpolation” is closer to the actual path of the mathematical function it approximates. Remember that we need to add all the vertical slices of the area: the more slices, the more terms to add together.
Rigourous Riemann Integrals
To attempt any rigour in progressing from these approximations, we need to use Bernhard Riemann’s work on “Riemann Integrals”.
To Define A Finite Partition of Real Numbers
Riemann’s work starts with a “finite partition” in the set of real numbers. Let’s set “x(0) = a < x(1)…< x(n) = b”. Then we can create a partition of [a, b] as a set of ‘n’ sub-intervals [x(0), x(1)], [x(1), x(2)],… [x(n-1), x(n)]. The “n+1″ points form a partition, ‘P’, and we can say P = {x(0), x(1),…x(n)}.
Each subinterval [x(i), x(i+1)] has the length L = x(i+1) – x(i). The largest length of a subinterval of ‘P’ is called the “norm of P”, and is shown as “||P|| = maximum for all ‘i’ of ( x(i+1) – x(i) )”.
To Define a Refinement of a Partition
We will need to “refine a partition”, or slice it thinner. Say that P’ is a partition over the same [a, b] as partition P, with more points x’(i). If every point x(i) in the original partition P is also a point in P’, but there are additional points in P’, then P’ is a refinement of P.
To Define a Riemann Sum of a Partition
In case the web site does not support the “less than or equal” sign, it is shown as “<=”.
Let f(x) be a function, ‘f’, defined for all points in a partition P. Let {c(i)} be a set of real numbers in the partition P, such that x(i) <= c(i) <= x(i+1).
The Riemann sum, σ (“sigma”) of this function is the sum from j = 0 through (n-1) of f( c(j) )*( x(j+1) – x(j) ).
"Image of Riemann Sum Formula" Image by Mike DeHaan
It is important to realize that the c(j) values may take any value from x(j) to x(j+1), so there is an infinite number of values for a Riemann sum.
Return of the “Delta-Epsilon Limit”
As in the previous article on differential calculus, we will use “delta” (‘δ’) to mean a small change in ‘x’, and “epsilon” (‘ε’) to mean the change in the function f(x) over that δ range.
Let f(x) be a function defined on the real numbers in partition P of [a, b]. Let σ be any Rieman sum of function ‘f’ on P. Let δ and ε be small real numbers, greater than zero. Let L be a real number, such that the absolute value |σ – L| < ε when the maximum subinterval ||P|| < δ.
When all these conditions are met, ‘L’ is the Riemann integral of the function ‘f’ over the partition ‘P’ from [a, b]. The symbol for “integral” follows.
"Image of Parabola Showing Exact Area Under the Curve" by Mike DeHaan
The Simplest Example of a Riemann Integral
If “f(x) = a” where ‘a’ is a constant real number, then any Riemann sum σ of “f = sum of( a*( x(j+1) – x(j) ) )”. That is simply “a*( ( x(n) – x(n-1) ) + ( x(n-1) – x(n-2) ) +…+ ( x(2) – x(1) ) + ( x(1) – x(0) ) )”.
But the right side has many cancelling pairs of “… – x(n-1) + x(n-1)…”, so it leaves “a*( ( x(n) – x(0) ) )”.
Amazingly, this is exactly what we found in the first examples where the rate was a constant value, and the start (now shown as “x(0)”) was zero.
This article needs to close before examining more complicated integration.
Closing Notes for the Riemann Integral
Mathematicians would add that this Riemann integral from ‘a’ to ‘b’ exists if and only if the function ‘f’ is “Riemann integrable” on the partition [a, b].
That should serve as a warning: some functions cannot be integrated. Remember that it is impossible to find the differential of some functions, too. A typical example would be “y = 1/x”, where (0,?) is undefined.
Who was Bernhard Riemann?
Bernhard Riemann: Image by Ævar Arnfjörð Bjarmason
Bernhard Riemann was born in Hannover in 1826. After graduating from Göttingen University, he began studying theology but transferred to philosophy in order to pursue his long-evident mathematical gifts.
After brief studies elsewhere, Riemann returned to Göttingen to complete his career. He addressed a number of important issues, including the introduction of an “n-dimensional Riemannian manifold” that Albert Einstein would use, sixty years in the future, in his theory of general relativity.
His greatest work might be the still-unproven conjecture called the Riemann hypothesis. It has significant implications for the distribution of prime numbers.
Riemann married in July of 1862. Later that year, a common cold developed into tuberculosis. He died in Italy in 1866, survived by his wife and a daughter, aged three.
"Integral Symbol" Image by M. DeHaan
References:
Hoffman,Mike. Bernhard Riemann. US Naval Academy. (Sept. 29, 2009). Accessed Aug. 6, 2011.
Trench, William F. Introduction to Real Analysis. Trinity University, San Antonio. (2003). PDF accessed Aug. 6, 2011.
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