5 4 Notes


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5 4 Notes

  1. 1. 5.4 Area and the Definite IntegralGoal: To find the area under a curve<br />
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  5. 5. As we break our graph into more and more rectangles, we get a better approximation of the area under the curve. We just continue to add the areas of each of the smaller rectangles.<br />Let’s say our graph has n thin strips each of identical width.<br />So we can say that each strip is n wide by f(x) tall.<br />
  6. 6. We can think of n as just Δx since it is the just the amount by which the x value changes and it is the same for every rectangle. <br />We can use c1c2,c3… and so on for each of the x values that we are substituting in, in order to calculate f(x).<br />So the area for the first rectangle is Area = f(c1)Δx, for the second Area = f(c2)Δx and so on<br />If we carry this out all the way we can say that:<br />Which means:<br />Or the sum of all the areas of the rectangles<br />
  7. 7. Since more rectangles = more accuracy having ∞ rectangles would be best.<br />The limit was first solved without uniform continuity by Cauchy in 1823, it was later proven and put on a solid logical foundation by Georg Friedrick Bernhard Riemann (1823-1866)<br />His work led to the following definition for definite integrals:<br /> If f is continuous on [a,b]<br />Exists and is the same number for any choice of ck<br />
  8. 8. The Midpoint Rule: If we cannot integrate the function for which we need to find the area under the curve, we can use Riemann sums (the midpoint rule)<br />Basically, we figure out how many rectangles we need. Find the width and then multiply by the sum of the lengths.<br />
  9. 9. Use the Midpoint Rule with n = 4 to approximate:<br />First break [0,1] into 4 intervals and find the midpoint of each.<br />0 - ¼ ¼ - ½ ½ - ¾ ¾ - 1 Δx = ¼ <br /> 1/8 3/8 5/8 7/8 are the midpoints for each interval<br />≈ .4358<br />
  10. 10. Use the Midpoint Rule with n = 4 to approximate:<br />First break [2,4] into 4 intervals and find the midpoint of each.<br />2 – 2.5 2.5-3 3-3.5 3.5-4 Δx = ½ <br /> 9/4 11/4 13/4 15/4 are the midpoints for each interval<br />≈ 5.6419<br />
  11. 11. The definite integral in terms of area under the curve:<br /> Let f be non-negative and continuous on the closed interval [a,b]. The area of the region bounded by the graph of f, the x-axis and the lines x = a and y = b is denoted by <br />Newton and Leibniz are both credited with achieving this great development in calculus. Newton derived it first, but Leibniz published first.<br />
  12. 12. EX<br />
  13. 13. EX<br />
  14. 14. EX<br />
  15. 15. Average Value of a function or Mean Value Theorem:<br />If f is continuous on [a,b] then the average value of f on [a,b] is <br />
  16. 16. Even and Odd Functions<br />If a function is symmetric with respect to the y-axis it is called an even function.<br /> so f(-x) = f(x) The value of the function at x and –x is the same<br />If a function is symmetric with respect to the origin it is called an odd function.<br /> so f(-x) = -f(x) The value of the function at -x is the opposite value of the function evaluated at f(x).<br />
  17. 17. For even functions:<br />For odd functions:<br />
  18. 18. EX<br />Vs.<br />
  19. 19. The Calculator!<br /><ul><li>fnInt(function,variable,lower bound, upper bound)
  20. 20. fnInt is found under math num 9</li></ul>fnInt((2x+3)^3,x,1,4)=<br />1752<br />