http://www.ck12.org Chapter 4. Integration
∫ 3
0 x(^3) dx=^81
4.
Before we look to try some problems, let’s make a couple of observations. First, we will soon not need to rely on
the summation formula and Riemann Sums for actual computation of definite integrals. We will develop several
computational strategies in order to solve a variety of problems that come up. Second, the idea of definite integrals
as approximating the area under a curve can be a bit confusing since we may sometimes get results that do not make
sense when interpreted as areas. For example, if we were to compute the definite integral∫−^33 x^3 dx,then due to the
symmetry off(x) =x^3 about the origin, we would find that∫−^33 x^3 dx= 0 .This is because for every sample pointx∗j,
we also have−x∗jis also a sample point withf(−x∗j) =−f(x∗j).Hence, it is more accurate to say that∫−^33 x^3 dxgives
us thenet areabetweenx=−3 andx= 3 .If we wanted thetotal areabounded by the graph and thex−axis, then
we would compute 2∫ 03 x^3 dx=^812.
Lesson Summary
- We used Riemann Sums to approximate areas under curves.
- We evaluated definite integrals as limits of Riemann Sums.
Multimedia Link
For video presentations on calculating definite integrals using Riemann Sums(13.0), see Riemann Sums, Part 1
(6:15)
MEDIA
Click image to the left for use the URL below.
URL: http://www.ck12.org/flx/render/embeddedobject/563and Riemann Sums, Part 2 (8:31).
MEDIA
Click image to the left for use the URL below.
URL: http://www.ck12.org/flx/render/embeddedobject/564The following applet lets you explore Riemann Sums of any function. You can change the bounds and the number
of partitions. Follow the examples given on the page, and then use the applet to explore on your own. Riemann
Sums Applet. Note: On this page the author uses Left- and Right- hand sums. These are similar to the sums
S(P)andT(P)that you have learned, particularly in the case of an increasing (or decreasing) function. Left-hand