Math 1210 ‐ Section 5.1 ‐ Areas and Distances
For the most part in this course, we have been dealing with the rates of change of functions. This is the branch of calculus called differential calculus. We now want to look at how functions accumulate the quantities they measure. This branch is called _______________ calculus.
To start, we’ll look at the way area accumulates for a very irregular shape, your hand.
Example 1‐ Now, suppose we travel for 5 hours at 60 miles per hour. What distance would we travel in those 5 hours?
How about from hour 3 to hour 5?
The velocity function (which measures the r.o.c. of _______________ ) has accumulated a change in position. In addition to solving this problem with the simple calculation above, we can also solve it by looking at the graph of the velocity function and answering some basic questions about the rectangular shape under the velocity function.
What if the velocity wasn’t constant? Would it still work the same way? Or, would we need to make adjustments?
Example 2‐ Suppose and we traveled for 3 hours.
In the example 1, we used a geometric formula. This can be very effective in finding the area between the curve and the horizontal axis but only if it’s a familiar geometric shape. (e.g.‐ rectangle, triangle, circle, trapezoid, etc.)
There are several basic approaches to find the area between a function and the horizontal axis, all of them start the same way:
(in mph)
(in hrs)
1. Partition the interval into subintervals.
equal subintervals on the interval , can be found using the formula
∆ This is called a ________________ partition.
2. Make rectangles that approximate the area on each subinterval using as the _____________ and ∆ as the ____________ of each “rectangular slice”.
3. Add the area of the rectangles to approximate the area between the curve and the axis.
(Finding areas this way is called a Riemann Sum, named after Georg Bernhard Riemann.)
What type of graph is this?
How can we find this area?
How tall is the rectangle? Units?
How wide is the rectangle? Units?
What is the area of the rectangle?
*The area between the velocity curve and the
horizontal axis is the ______________ change
accumulated during the time interval.*
(i.e. It’s the displacement. Displacement is
the ________ change in position.
What type of graph is this? ____________ What could we do to improve the accuracy
Right Riemann Sum Left Riemann Sum Midpoint Riemann Sum
Find each area approximation above using · ∆ for the area of each rectangular slice.
Right Sum= Left Sum= Mid‐pt. Sum=
Which is an upper sum (overestimate)? Which is a lower sum (underestimate)?
Is a left sum always a lower sum for every graph? What type of graph would make a left sum an upper sum?
Is there a shape that would be better than a rectangle?
Regardless of how we choose the heights of the rectangles, how could we partition the interval to make all of these approximations more accurate?
Example 3‐ Find an approximation for the area between sin and the axis from 0 to . Make a regular partition with 4 subintervals and do a Left, Right, or Midpoint Riemann Sum. (Or, you could choose the heights so the rectangles are always an upper sum or a lower sum. This would be a method in addition to the five described above.)
(You can see that ___________________
work very well.)
In section 5.2 we’ll see that you can partition any way you want and choose any function value you want in each subinterval.
This idea is very powerful.
(Draw the rectangles using the ______
endpoint function value as the height.)
(Draw the rectangles using the _______
endpoint function value as the height.)
(Draw the rectangles using the ______‐
*Most of the problems in this section are fairly straightforward, but they do take time, patience, and accuracy. There is value in drawing the
rectangles, computing the areas, and adding them up. We're working our way to amazing application of this idea.*
Example 4‐ Suppose a car starts from rest and accelerates according to the following table. Approximately how fast is the car going after 5 seconds?
(sec) 0 1 2 3 4 5
(ft/sec2) 0 5 8 9 8 4
It would be nice to come up with a shorthand for these long addition problems. In Math 1050 you were introduced to _______________ Notation and the basic concepts, notation, and vocabulary we use when working with Series. This notation can be used to simplify adding the areas of the rectangular slices.
You can use any variable you like for the index. The most common choices are: , , , . This book prefers . Because we will be using sigma notation in this class and in Math 1220, it is well worth our time to review how this notation works. To begin with, there are finite series and infinite series.
Finite series‐ 1 2 3 4 5 (Finite number of terms.)
Infinite series‐ 2 2 2 2 2 (Infinite number of terms.) "Expanded Form"
Recall also that 2 4 6 8 10 can be condensed and written using sigma notation. _____ ___
___
Series that have finite sums can be added, subtracted, multiplied by a constant, and they still have a finite sum that’s pretty easy to find.
Example 5‐ Given ∑ 2 30 and 55 , then: a. 2 _____ b. 3 ______
Some common series have formulae that help us quickly get the sum. Consider the following examples.
Example 6‐ a. ∑ 4 b. ∑ 5 So,
∑
c
Example 7‐ Find the sum of the first 1000 integers. That is: ∑ First integers:
∑
Example 8‐ Use properties of sums to find ∑ 10 3.
1 2 3 4 5
−1 1 2 3 4 5 6 7 8 9
If you need practice with sigma notation, look at the examples and exercises from Appendix E.
What type of graph is this?
_____________________
Summand‐ the formula that produces the terms you add.
In our case, the summand is the height times the width of each rectangle. Index‐ tells us to start with .
Greek letter Sigma‐ tells us to add.
Some other formulae that may be helpful:
First squares:
First cubes:
While it makes perfect sense that a finite series would have a sum (since there are only a finite number of terms to add up), it is also possible for infinite series to have a sum.
Example 9‐ Do the following infinite series have a sums? Why or why not?
a. b.
Now, let’s get back to adding rectangular slices. In order to increase the accuracy of our rectangular approximations, we need to formalize how we compute and add the areas. Using sigma notation to write Riemann Sums is part of this formalization.
Example 10‐ Estimate the area between the curve and the axis on 0 , 3 by partitioning the interval into 3 equal subintervals. Use subscripts to distinguish the evaluation point of each subinterval. Write the sums using sigma notation using the right, left, and mid‐points.
Now, set up a sum using sigma notation that would calculate the area using 12 rectangles and right endpoints.
1 2 3
1 2 3 4 5 6 7 8 9
We'll use to represent the point at which we
evaluate the height of the function in the
subinterval. If is an endpoint or mid‐point,
then we use:
Right Endpoint: · ∆
Left Endpoint: · ∆
Mid‐point: · ∆
How about 300 rectangles?
Trade problem 13 for 18 on the homework assignment.
As discussed earlier, we can make any Riemann Sum more accurate by using more rectangles. Consider the following:
Example 10*‐ Estimate the area between the curve and the axis on 0 , 3 by partitioning the interval into _____ equal subintervals and following the method outlined above.
Since we have generalized this approach for rectangular "slices", what would stop us from using as many rectangles as we want? How many slices would make this estimate perfect? ___________
*This suggests that to calculate the exact area, we will need to take the
____________ of this rectangular "slicing and summing" process as _______.*
Example 10**‐ Find the area between the curve and the axis on 0 , 3 by partitioning the interval into equal subintervals and following the method outlined above. Then find the limit as ∞.
Example 3*‐ Set up a limit to find the area between sin and the axis from 0 , using right endpoints.
The textbook refers to the method above as definition 2.
Example 11‐ Find an expression for the area between the curve √ and the axis on 5 , 9 using definition 2. Do not find the limit.
Can you identify the · ∆ and the interval on which we are accumulating area for the previous problem?
Why have we been using the right endpoint in most of the examples?
*OPTIONAL*
Finding areas using a large number of slices can be very difficult and finding limits of the above problems can be extremely difficult or even impossible. However, we can use technology to make estimating areas easier. The process below outlines how you can use a TI‐84 to estimate areas using a given number of rectangular slices.
Enter your function into “ ”. Then from the home screen you’ll need to access the “LIST” menu (that's the 2nd function of the “STAT” button) to enter the following syntax:
∆ , , ,
Where is given by a formula above, ∆ , is the index, and is the number of subintervals. (We type letters by hitting the “ALPHA” button then the letter you want.)
For a mid‐point Riemann Sum we would have: . ∆ ∆ , , ,
You can save yourself a lot of typing and editing if you know how to store variables. You can store variables by hitting the “STO ” button (just above the “ON” button). To store the left endpoint in , enter the left endpoint, then hit “STO ”, then “ALPHA”, then , then “ENTER”. Your screen will show this, , and the left endpoint is now stored in . You’ll need all of the following to enter any Riemann sum into the calculator quickly by simply changing , , ,
, and hence .
, , , / , and , , 1,
Once these commands are entered, they can be recalled and edited, making multiple calculations with the same or different functions and/or intervals relatively easy.
Example 12‐ Use your calculator to find the area between the curve √1 and the axis on 0 , 1 . Try 50, then 100, then 1000. (There is a certain satisfaction that comes from overwhelming a calculator)