We compute Riemann Sums to approximate the area under a curve.
The first major problem in calculus was the tangent line problem. We now turn to the second major problem in calculus, the area problem. We saw that the derivative solved the tangent line problem and it turns out that the anti-derivative solves the area problem.
The Area Problem
Let be a non-negative, continuous function on the interval . The area problem asks us to find the area under the graph of over the interval .
To solve this problem, we begin by approximating the area under the curve using rectangles. The sum of the areas of these rectangles is called a Riemann Sum.
To find the exact area under the curve we will need to use infinitely many rectangles. This will lead us into the next section on the Definite Integral. For now, we begin with a special notation that will allow us to write the sum of the areas of many rectangles in a compact form.
Summation Notation
In mathematics, the symbol is used to denote summation.
Summation Notation The
parameter is called the index of summation and it begins at the indicated value,
in this case; and it increments by one unit until it reaches the terminal value (in this
case, ).
Special Sums Sums of powers of :
- 1)
- The sum of the first whole numbers:
- 2)
- The sum of the first perfect squares:
- 3)
- The sum of the first perfect cubes:
example 3 Find the value of the sum: This sum can be found by rewriting it as a
difference of two sums and using formula (2), above, twice:
Riemann Sums
Let be defined on a closed interval . Partition the interval into sub-intervals, each of width Denote the endpoints of the sub-intervals by , so that where . Note that and
To create rectangles under the curve, we next select sample points- one in each sub-interval so as to space them out roughly evenly over the interval . The sample points are denoted by an asterisk, like for the fourth sample point and for the -th sample point.
In general, the -th sample point, lies in the interval . The sample points are used to determine the heights of the rectangles used in our Riemann Sum. Specifically, the height of the first rectangle will be , the height of the second rectangle will be and so on until we reach the -th and final rectangle, whose height will be .
For a function which is positive on the interval the Riemann Sum represents
the sum of the areas of rectangles. The heights of the rectangles are given
by the function values at the sample points, , and the widths are given by
.
example 4 Below is an interactive graph of the parabola . The Riemann
Sum uses the rectangles in the figure to approximate the area under the
curve. The sample points are taken to be endpoints of the sub-interval . The
orange rectangles use , i.e., a left-endpoint approximation and the purple
rectangles use a right-endpoint approximation with . Use the slider to
convince yourself that as the number of rectangles, increases, the width
decreases and the area of the rectangles approaches the area under the curve.
example 5
The value of this Left Riemann Sum is which equals
Find the Riemann Sum for the curve over the interval using rectangles and left endpoints as the sample points.
(problem 5) Find the Riemann Sum for the curve over the interval using rectangles
and left endpoints as the sample points.
The Riemann Sum is .
