MATH 149LABORATORY ASSIGNMENT 7THE DEFINITE INTEGRALIn class we defined d⌠⌡⎮⎮ab()f xx as Lim∑ = k 0 − n 1()f ck() − x + k 1xk , where a=x0<x1<x2<...<x − n 1<xn=b is a partition of [ a , b ] , xk<ck<x + k 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 X-axis, and the lines x = a and x = b. The sum ∑ = k 0 − n 1()f ck() − x + k 1xk represents the total area of the n rectangles with base [ xk , x + k 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 , x + k 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 RECTANGLESThe 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)));1.416241251The 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 → n ∞An.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));15⎛⎝⎜⎜⎞⎠⎟⎟sin11015⎛⎝⎜⎜⎞⎠⎟⎟sin31015⎛⎝⎜⎜⎞⎠⎟⎟sin1215⎛⎝⎜⎜⎞⎠⎟⎟sin71015⎛⎝⎜⎜⎞⎠⎟⎟sin91015⎛⎝⎜⎜⎞⎠⎟⎟sin111015⎛⎝⎜⎜⎞⎠⎟⎟sin131015⎛⎝⎜⎜⎞⎠⎟⎟sin32 + + + + + + + 15⎛⎝⎜⎜⎞⎠⎟⎟sin171015⎛⎝⎜⎜⎞⎠⎟⎟sin1910 + + Using the "Sigma" notation, the last expression can be written as:> A[10]:=middlesum(sin(x),x=0..2,10); := A1015⎛⎝⎜⎜⎜⎞⎠⎟⎟⎟∑ = i 09⎛⎝⎜⎜⎞⎠⎟⎟sin + i5110In general, if we construct n boxes and calculate their total area, we get> A[n]:=middlesum(sin(x),x=0..2,n); := An2⎛⎝⎜⎜⎜⎜⎜⎞⎠⎟⎟⎟⎟⎟∑ = i 0 − n 1⎛⎝⎜⎜⎜⎜⎞⎠⎟⎟⎟⎟sin2⎛⎝⎜⎜⎞⎠⎟⎟ + i12nnMaple knows a formula for this sum:> value(%);2⎛⎝⎜⎜⎜⎜⎜⎜⎜⎞⎠⎟⎟⎟⎟⎟⎟⎟ − ⎛⎝⎜⎜⎞⎠⎟⎟sin1n()cos 12 − ⎛⎝⎜⎜⎞⎠⎟⎟cos1n21⎛⎝⎜⎜⎞⎠⎟⎟sin1n − ⎛⎝⎜⎜⎞⎠⎟⎟cos1n21nThe 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 122To 10 decimal places, the area is:> evalf(%);1.416146836> Thus, to 10 decimal places, d⌠⌡⎮⎮02()sin xx = 1.416146836.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 7Section: <your section goes here>> restart;Exercises 1. Let f (x) = + 8 x37 , 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 < π24. . The exact area under the curve is − π 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
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