Section 4.3 Riemann Sums and Definite Integrals

To this point, anytime that we have used the integral symbol we have used it without any upper or lower boundaries. In other words, it has served as simply a short hand symbol asking you to antidifferentiate a function. We have also been talking about areas simultaneously. Those problems do have upper and lower bound limits. There is a reason that we have been pursuing these two seemingly different goals at the same time.

Section 4.3 Riemann Sums and Definite Integrals The reason why is that they are the same goal, not different ones. Using words to describe a problem:  Find the area of the region bounded by the curve,the x -axis, and the lines x = 2 and x = 6. Using summation to describe the problem Using integral notation to describe the problem

Section 4.3 Riemann Sums and Definite Integrals Below is a rough approximation of the region bounded by a curve, just call it f(x ) and the vertical lines x = 2 and x = 4. Write this as a definite integral.

Section 4.3 Riemann Sums and Definite Integrals Draw the region defined by the integral below:

Section 4.3 Riemann Sums and Definite Integrals Did your graph look like the one below?

Section 4.3 Riemann Sums and Definite Integrals Now try to read a summation notation and turn it into a picture. Here’s the sigma notation:

Section 4.3 Riemann Sums and Definite Integrals Does your picture look like this?

Section 4.3 Riemann Sums and Definite Integrals Now, let’s untangle some of the notation in the text for this section. There is a discussion of Riemann Sums that seems to be related to the integral. The notation in the definition box on page 272 certainly looks like an area formula. What does not look familiar is the notation below it. The breakthrough that Georg Riemann had was that regions do not have to be subdivided into equal base segments to sum their areas as we have been doing. This is an important historical piece, but it will not change how or what we do when calculating these areas.

Section 4.3 Riemann Sums and Definite Integrals The final piece in this section is the presentation of important properties of integrals on pages 276 – 278  The integral of a region can be found by adding integrals for adjacent, non-overlapping regions. This can be helpful when looking at piecewise defined functions.  The integral of a sum or difference of functions is the sum or difference of the integrals of the functions.  The integral when evaluated from a to b is the opposite of the integral when evaluated from b to a.  If a curve is above the x -axis on a region, its integral on that region is positive.  If a function f is less than a function g on an interval, then the integral of f is less than the integral of g on that interval