1. Estimating Area of Shapes. We can consider limits of finite sums when estimating the areas of shapes in the plane. So for example, the triangle:
describes the triangle’s hypoteneuse.
What can we do? Well, we can pick a number of points . The points are equidistant from each other, so
Then we form a number of rectangles:
We have shaded the area of the rectangles. Note there is some defecit area. What is the area of one of these rectangles? It is
Pop Quiz: is ? ? ? ?
The reader may verify these describe the rectangle’s area. Thus
Thus we find the area of all the rectangles by
Factoring out , which is constant with respect to the dummy index , we get
Using the sum rule yields
The sum transforms our problem
Using Gauss’ sum we find
Basic arithmetic yields
So we get
What’s the error of this approximation? It’s the unshaded regions of the triangle, which is about .
We can refine the approximation, taking twice as many points. So this changes . The approximation gets better, but still only approximates the area. We can take the limit
This gives us the area of the triangle.
2. Riemann Summation. We generalize this scheme approximating the area between the curve and the axis. We consider a closed interval . We choose an arbitrary partition of the interval, i.e., a set of points . They don’t have to be equally spaced. Notice this consists of closed subintervals! There are intervals and the width of the -th interval is
The Riemann sum approximating the area is
How do we get the area? We take the limit . This gives us the “Riemann Integral”. This is the definite integral
Note if is the antiderivative of , then
is the definite integral from to .
In a sense, what happens is our limit makes the following transformation
The sum is transformed into a “continuous sum” (i.e., integral) .
So we can keep the intuition of adding up a bunch of rectangles, but now the rectangles have width (an “infinitesimally” thin rectangle!) and height .
Of course, infinitesimals do not exist. But it is a useful intuitive picture to have. The idea that invokes fond memories of the derivative. We will show in future posts (viz. the Fundamental Theorem of Calculus) this is no coincidence!
Exercise 1. Consider the Riemann sum for on the domain . What is its limit as the number of intervals ? [Hint: see exercise 1 of the finite series post.]
Exercise 2. Consider the Riemann sum for on the domain . What is its limit as the number of intervals ? [Hint: see exercise 2 of the finite series post.]
Exercise 3. Consider the Riemann sum for on the domain . What is its limit as the number of intervals ?