MATH 149 LABORATORY ASSIGNMENT 7 THE DEFINITE INTEGRAL b n 1 In class we defined f x dx as Lim a f c x k k 1 xk where a x0 x1 x2 xn 1 xn k 0 b is a partition of a b xk ck xk 1 and the limit is taken as n and the lengths of the subintervals determined by the partition go to 0 If f x 0 for all x in a b then the integral represents the area of the region bounded by the curve y f x the n 1 X axis and the lines x a and x b The sum f c x k k 1 xk represents the total k 0 area of the n rectangles with base xk xk 1 and height f ck k 0 n 1 There are commands in MAPLE s student package that allow us to easily draw the approximating rectangles and calculate the corresponding sums for partitions in which the xk s are evenly spaced and the ck s are either the left hand endpoints the right endpoints or the midpoints of the intervals xk xk 1 determined by the partition As an example we will let f x sin x a 0 and b 2 restart with plots Warning the name changecoords has been redefined with plottools Warning the assigned name arrow now has a global binding A plot sin 0 2 B line 2 0 2 sin 2 color red C textplot 1 2 0 4 REGION R color red E textplot 0 7 0 8 y sin x display A B C E We will initially approximate the area of the region using 5 rectangles the partition points will be equally spaced and the ck s will be the midpoints of the subintervals thus the partition points are 0 0 4 0 8 1 2 1 6 and 2 and the ck s are 0 2 0 6 1 0 1 4 and 1 8 with student middlebox sin x x 0 2 5 We now increase the number of rectangles middlebox sin x x 0 2 10 10 RECTANGLES middlebox sin x x 0 2 20 20 RECTANGLES middlebox sin x x 0 2 50 50 RECTANGLES The total area of the rectangles above can be calculated using Maple evalf value middlesum sin x x 0 2 5 1 425632060 evalf value middlesum sin x x 0 2 10 1 418509838 evalf value middlesum sin x x 0 2 20 1 416737070 evalf value middlesum sin x x 0 2 50 1 416241251 The exact value of the area of the region R will be found by calculating An the total area of n rectangles constructed as above and then finding lim An n As an example we will show how A10 is calculated middlebox sin x x 0 2 10 Each of the 10 boxes has width 1 5 2 units divided into 10 parts The height of the first box is sin 1 10 the height of the second is sin 3 10 the third sin 5 10 etc Therefore the total area of the boxes is value middlesum sin x x 0 2 10 1 1 1 3 1 1 1 7 1 9 1 11 1 13 1 3 sin sin sin sin sin sin sin sin 5 10 5 10 5 2 5 10 5 10 5 10 5 10 5 2 1 17 1 19 sin sin 5 10 5 10 Using the Sigma notation the last expression can be written as A 10 middlesum sin x x 0 2 10 9 1 i 1 A10 sin 5 i 0 5 10 In general if we construct n boxes and calculate their total area we get A n middlesum sin x x 0 2 n 1 n 1 2 i 2 sin 2 n i 0 An n Maple knows a formula for this sum value 1 1 sin cos 1 2 sin n n 2 2 2 1 1 cos 1 cos 1 n n n The area can now be found by taking the limit as n goes to infinity which Maple also knows how to do limit n infinity 2 cos 1 2 2 To 10 decimal places the area is evalf 1 416146836 2 Thus to 10 decimal places sin x dx 1 416146836 0 Evaluating such sums and limits directly as in the above example is sometimes beyond Maple s capability However the required area can always be approximated to any number of decimal places by calculating An for a large enough choice of n The question however is exactly how large must you take n in order to be guaranteed accuracy to the desired number of decimal places This is a non trivial question which is addressed in advanced mathematics courses on Numerical Analysis All we can do at this stage is investigate using examples Name your name goes here Lab 7 Section your section goes here restart Exercises 3 1 Let f x 8 x 7 1 x 3 The exact area under the curve is 164 a Use leftbox and rightbox with 20 rectangles each to plot rectangular approximations to the area under the curve Which approximation is too large and which is too small Why b Use leftsum and rightsum to calculate approximate values for the area c Repeat the calculations in part b with n 40 and 80 How do the errors in these approximations change as n increases d Find the smallest value of n for which the error in in using leftsum is less than 0 000005 Roundoff errors may play a role You may wish to increase the number of digits e Use MAPLE to find a formula for the leftsum and the rightsum with n terms f Use MAPLE to find the limit of these sums as n 2 Let g x cos x 0 x 2 4 The exact area under the curve is 2 a Find the smallest value of n such that the middlesum approximation is accurate to 4 decimal places b Try to use MAPLE to evaluate the integral as in e and f above Use middlesum
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