Think about the space between the graph of the function f(x) = x2, the x-axis, the y-axis, and the vertical line x = 1. The area of this roughly triangular shape is a bit less than one half. By following the steps below, we can use a computer to get a much better approximation of its area than just "less than one half". The way we do this is to fill the triangular shape with tall, skinny rectangles then add the areas of the rectangles. This total is called a Riemann Sum.
Take a second to think about your results. The numbers listed in column C are the areas of rectangles that fit under the graph of the function f(x) = x2. This means that the area under the graph of the function is probably just a little bit more than the sum of those areas because some of the space under the curve was not covered by rectangles.
"A little more than 0.24" is a much better estimate than "less than one half". You can improve this answer still further by making your rectangles thinner. To compute a new estimate using areas of rectangles with width .05 units, type = A2 + .05 in cell A3 and Fill down to cell A22, then fill columns B and C down to cells B22 and C21 and set cell D2 to = SUM(C2:C21).
Question 1: What estimate of the area under the graph of f(x) = x2 do you get if you sum the areas of 20 rectangles with base width .05 units (as described above)?
Question 2: Your estimate in Question 1 should have been different from 0.24. Explain the difference between the two estimates. Which is more accurate? Why? Is the new estimate larger or smaller than the old estimate? Is it larger or smaller than the exact value of the area?
Question 3: What do you think the actual area is?
Question 4: If you have time, estimate the area again with even thinner rectangles. What base width did you use and what area estimate did you get?
Later, we'll be able to use calculus to find the exact area and check our work. However, for some functions (like the famous statistics function g(x) = e-x2) it is impossible to compute an exact integral! For functions like these you must use a tool like a calculator or spreadsheet to estimate the area. Those tools use methods very similar to our method of adding areas of rectangles.
Extra Credit: (Turn in a printout of your spreadsheet to add up to 1 point to your final grade.) What is the area between the graph of the function g(x) = e-x2, the x-axis, the y-axis, and the line x = 1? Note: most spreadsheets have a function EXP() that you can use to compute e to a power. For example, EXP(1) = e1 = 2.718...