Math 12 -- Calculus II
Santa Clara University
Department of Mathematics and
Computer Science
Dennis C. Smolarski,
S.J.
Math 12 Homepage (Smolarski)
Integration and Riemann Sums
The definition of the definite integral as a limit of sums
can be used to find the area "under" a curve.
In actuality, it finds the area under a positive function
and the x-axis,
and the "signed" area (that is, the difference
between the "positive" area above the x-axis
and the absolute value of the "negative" area below the
x-axis)
between an arbitrary curve and the x-axis.
The integral is used to accomplish this by approximating
the actual area by a sum of approximate areas.
These approximate areas are each rectangular-shaped
slices of the actual area.
We choose rectangles as our underlying figure
because of the simple formula for its area, i.e.,
Length times Width.
The width is represented as Delta_x and the
length corresponds to the height of the function at
some point x*i
along the graph (the various options are noted in the
next section), whose value is, therefore, merely
f(x*i). Thus the area of one of the rectangles
is f(x*i) × Delta_x. The
total approximate area is the sum of all the
f(x*i) × Delta_x for a specific
number of rectangles. The total exact area is the
limit value of this sum as the number of rectangles tends to infinity.
Shapes of Rectangles
There are many different ways to choose the
rectangular-shaped slices. We can use rectangles
that are inscribed (always completely within the actual area),
or that are circumscribed (always completely enclosing
the actual area),
or whose left edge matches the height of the curve, or whose
right edge matches the height of the curve, or whose
mid-point matches the height of the curve. We can also use
rectangles all of the same width (common practice) or of
different widths.
Approximate Area Leads to Actual Area
As noted above, as one increases n, the
number of rectangles, the rectangles of any width and
any height better and better approximate the actual area.
In the end, as n tends toward infinity, the
approximate area, represented by the sum of areas of the
rectangles (no matter what specific form),
and symbolized by the definite integral, tends
toward the exact area.
Example
In the following graphs depict the curve
y=sinx2.
![[Maple Plot]](images/reimanns2.gif)
If we drew five rectangles of equal width that
approximated the area, such that the
the left edge of each rectangle matched the curve, we
would get following figure:
![[Maple Plot]](images/reimanns3.gif)
If those rectangles were drawn such that the middle of the top (or
bottom) edge matched the curve, we would get the following figure:
![[Maple Plot]](images/reimanns4.gif)
As the number of rectangles increases, no matter which type are
used, they better and better approximate the exact areas
related to the curve.
Animated Example
In the following animation, the rectangles are such that the
left edge always lies on the curve,
y=sinx2. The animation shows
the number of rectangles increasing, with each step better approximating
the actual area between the curve and the x-axis, thereby
better and better approximating the exact area, which is given
by the integral
![[Maple Math]](images/intsin1.gif)
![[Maple Plot]](images/riemanns1.gif)
(NOTE: The animation can be stopped at any time by
pressing the escape key, and restarted by clicking on the Reload
button of the browser.)
Last Updated: 23 January 2023. Maintained by
Dennis C. Smolarski, S.J.