Math 1210 - Section 5.2 - The Definite Integral
Starter Questions- Can area ever be negative? What is the formula for the area of a trapezoid? Which is most accurate for approximations of area: Left, Right, or Mid-Point?
Why?
In the previous section, all of the function values were positive. How would you deal with the following problem? Example 1- Graph on , then find an approximation for the area between the curve and the
-axis with a Riemann sum like: . Use equal subintervals and let be the mid-point of
the subinterval. Remember: If is a mid-point, then
(To do this problem, let's proceed as normal and see if the answer makes any sense.)
This problem illustrates the concept of “Signed Area”. As we move from to (i.e. in the postive direction from ___________ to _____________), we accumulate area. Area above the -axis counts as ____________, area below the -axis counts as ____________. The “Net Area” is the sum of the signed areas (areas above minus areas below).
Because adding the signed areas accumulated by functions comes up so often, we give this process its own special name and notation. The process of accumulating signed area is called __________________. Here is the definition and notation used by this textbook:
There are several things we should note about this definition:
1. The notation and symbols for the definite integral seem a little strange. But, consider the following rationale for the notation of this partitioning/limiting process:
2. This definition uses equal subintervals, but this is not a requirement. As discussed in the previous section, since we’re taking the limit of this summing process as , the widths of all the rectangles will approach ______. So, it doesn’t matter if they have different widths or not, because they are all going to end up to be the same infinitely _____________ width.
x y
“____________________” Variable of Integration Integral Sign
____________ Limit of Integration
___________Limit of Integration
3- Continuous functions on a closed interval are Integrable. Theorem 3 takes this simple statement a bit further:
Example:
Now try, Example:
This last function is so discontinuous, that no area approximation can be found.
4. is a real ________________, it represents the net area between the curve and the horizontal axis. This net area depends only on the function (which makes the shape) and the upper and lower limits (which tells us where to start and stop adding area), it does NOT depend on the variable we use. This means we can use any variable we want as long as all the elements of the notation match.
Upper limit for ___
Lower limit for___ Function of ___ Differential of ___
To simplify the calculation required to integrate a function, we often use a regular partition and right endpoints.
This essentially says that the definite integral is the result of the same process we learned in the previous section. So, a thought question: If mid-points are more accurate, why don't we use them to find definite integrals?
We can now set up area problems as definite integrals and evaluate them using sigma notation.
Example 1*-Express as a definite integral on . Then use the limit to find the
net area between the curve and the -axis. (Does this look familiar?)
,
must be limits of ________.,
must be limits of ________.
Example 2- Set up an expression for each definite integral as a limit of sums:
a. b.
Example 3- Find the following definite integrals by interpreting them as areas.
a. b. c.
Example 4-Evaluate the following definite integrals by interpreting them as areas.
a. b. c.
When we approximated at the beginning of this section we had some function values that were negative. We accounted for the negative heights by saying that since area can’t be negative, we’ll call the product of the positive width and negative height of each rectangle the “signed area”. But consider this: the widths were all positive because we were going from to (we were going from _______ to _______). What adjustment would we need to make if we went from right to left? We can answer that question and many others by thinking geometrically.
Example:
Example:
Example:
Example:
Example:
We should make special note of the last property because it can be very useful. Notice that it’s an equation, and like any equation, we can use the rules of algebra to manipulate the equation. Use the following figure to find formulas for:
Example 5- Find the following based on areas given on the graph. (Pay close attention to the interval and direction.)
Basically, take the whole thing and subtract the piece you don't need.
We could also find:
a. b.
c. d.
e. f.
g. h.
The following properties are true only if we’re integrating from left to right (i.e. in the ______________ direction). Again, these are best understood by thinking about the rectangles that we could use to find the area.
Example 6- Use the last property above to give bounds for .
Example 7- How do the following compare?
a. and b. and
We’ll now look at how we can use geometry and sigma notation to help us find formulas for some common definite integrals.
Example 8- Find a formula for where , using the graph of and the formula for the area of a trapezoid.
Example 8*- Find a formula for where , using and
where is the right endpoint of the subinterval.
**The next section covers the most important theorem in this course. Please come prepared with answers to the following review questions from chapter 3:
1) What is the difference quotient for ?
