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# 7.4 - MATH 1081  Wednesday, April 13   ...

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Unformatted text preview: MATH 1081  Wednesday, April 13    Chapter 7 – Section 4    THE FUNDAMENTAL THEOREM OF  CALCULUS  Homework #12 (due 4/18):                Section 7.3 #8, 28, 36  Section 7.4 #4, 12, 26, 34, 56  Clicker Check­in:  Choose any letter to check in now.  Last time, we defined  b a n→ ∞   Geometrically, this represented the area of the region between  the function  f ( x )  and the  x ‐axis from  x = a  to  x = b .    We can approximate the value of this definite integral by using  some finite sum of areas of rectangles.  However, to find the  exact value we must evaluate the limit.    The Fundamental Theorem of Calculus tells us that actually this  limit is equal to the difference in the value of the  antiderivative of  f ( x )  at  x = a  and  x = b .  ⎣ ⎦ ∫ f ( x ) dx = lim ⎡ f ( x ) + f ( x ) + f ( x ) + … + f ( x )⎤ Δx .  1 2 3 n   So, we only need to find an antiderivative of  f ( x ) , evaluate it at  x = a  and  x = b , and then subtract (always the value at the  “top” bound minus the value at the “bottom” bound).    Note that we can use any antiderivative.  So, in particular, we  can chose the one where  C = 0  or whatever value we like.    Why is the  C  unimportant here?    Example:  Evaluate the definite integral.    4⎛ 12 ⎞       1.   ∫ ⎜ − x + 4 ⎟ dx   0⎝ ⎠ 4       31       2.   ∫ dx   1x       1 e2 x       3.   ∫ dx   2x 0 1+ e   In the last clicker question, we found the value of the definite  integral to be negative.  If this is meant to represent the “area  under the curve”, how can it be negative?    In this case, we were considering an interval where the  function was actually below the  x ‐axis.  So, in the sum  ⎡ f ( x1 ) + f ( x2 ) + f ( x3 ) + … + f ( xn ) ⎤ Δx , every  f ( x* ) is negative,  ⎣ ⎦ making the result negative.      In fact, a better interpretation of the definite integral is the  signed area or net area of the region between the curve and the  x ‐axis.  Regions above the  x ‐axis contribute a positive amount  and regions below it contribute a negative amount.  In general, we want the definite integral to represent the  signed (or net) area.  Thinking in terms of a velocity function, a  negative velocity would imply that we are travelling in reverse  somehow.   So, when computing a total distance, we want the  “backward” moments to be subtracted from the “forward”  moments.    In some cases, you will be asked to find the total area between  the curve and the  x ‐axis.  In this case, we do NOT want those  regions below the  x ‐axis to deduct from the total area.  We  want everything to be positive.    Example:  Determine the total area     1. of the shaded region.      2. between the  x ‐axis and the curve  f ( x ) = e x − 1 on   the interval [ −1, 2 ]     Note that the Fundamental Theorem of Calculus gives us a  method to evaluate the definite integral.  It does not define it.   The definite integral is defined to be the limit of a sum.    This is similar to how the rulesof differentiation simply give us  a method to find a derivative.  However, a derivative is defined  as a limit of a difference quotient.      Clicker Check­out:  Choose any letter to check out.    Tomorrow in recitation:  Quiz on Sections 9.2, 9.3, 7.1, 7.2    Next time – Section 7.5      ...
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