Unformatted text preview: Section 7.7 This section deals with techniques for approximating the value of definite integrals. We know we can determine the value of a definite integral by determining an antiderivative of . However, not all functions have antiderivatives, and also sometimes antiderivatives are just too difficult to find. Thus to approximate values of definite integrals, several techniques have evolved. These all involve the geometric aspect of area. Additionally, since these calculations are approximations, in applications it is important to determine the "worst case" amount of error in the approximation. Thus we have theorems that guarantee the error to be less than a certain amount. We already can approximate definite integrals using areas of rectangles -- that's what Riemann Sums are. As it happens, Riemann Sums using the midpoint rule are very good approximators of definite integrals. In this section we use areas of trapezoids and areas under parabolic arcs to approximate definite integrals. The Midpoint Rule: , where and Thm 4.16 The Trapezoidal Rule Let be continuous on is given by The Trapezoidal Rule for approximating Moreover, as , the right hand side approaches The trapezoidal rule "replaces" segments of the graph with line segments, forming trapezoids. The trapezoidal rule just adds together the areas of those trapezoids. MTH 173 section 7.7, page 1 Thm 4.18 Simpson's Rule ( is even) Let be continuous on given by Simpson's Rule for approximating is Moreover, as , the right hand side approaches Simpson's rule "replaces" segments of the curve with parabolic arcs and adds together these areas. Thm 4.19 Upper bounds on errors If has a continuous second derivative on , then the error approximating by the Trapezoidal Rule is max Moreover, if has a continuous fourth derivative on approximating by Simpson's Rule is max The maximums of these second and fourth derivatives can be estimated from the graph of these functions. , then the error in in MTH 173 section 7.7, page 2 Example: ! Midpoints: " " "" Trapezoidal rule:
! " MTH 173 section 7.7, page 3 Simpson's Rule: ! And by FTC2, the exact value MTH 173 section 7.7, page 4 Example sin
# ! sin sin sin
! ! " sin ! $ sin Trapezoidal rule:
$ " $ Simpson's Rule: " sin
# $ !", using Derive MTH 173 section 7.7, page 5 Example, given Estimate maximum error using trapezoidal & Simpson's rules. Trapezoidal rule: so we graph in max and estimate its max abs: % in , so "$ Simpson's rule: "$ & max in and estimate its max abs: so we graph the fourth derivative of % so $' MTH 173 section 7.7, page 6 Example Determine necessary number of terms so that From above, we know the max values of 24, so a. trapeziodal rule: terms b. Simpson's rule: and in using in the interval are 2 and % ( $ % ( % ( % ( MTH 173 section 7.7, page 7 ...
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