Nme Student ID # Instructor L Period Dte Due L 11 Approximte Integrtion Ojectives 1. To ecome fmilir with the right endpoint rule, the trpezoidl rule, nd Simpson's rule. 2. To compre nd contrst the properties of these three methods of pproximte integrtion. Explortion 1 Right Endpoint Rule The vlue of the definite integrl endpoint rule f (x)dx cn e pproximted using the right where x = ( )/n nd x i = + i x. n f (x)dx R n = f (x i ) x = x f (x i ) i =1 In this explortion, we will use the right endpoint rule to pproximte the vlue of the known definite integrl, following: n i = 1 sin( x)dx =1. But first, we must set up TEMATH y doing the Select New Constnt from the Work menu. Delete the defult constnt nme nd enter =. Select New Constnt from the Work menu. Delete the defult constnt nme nd enter = (press the Option p key for ). Select New Function from the Work menu. Delete the defult function nme nd enter f(x) = sin(x). Select New Function from the Work menu. Delete the defult function nme nd enter the right endpoint rule (press the Option w key for ) R(n) = ( )/n (i, 1, n, f(+i( )/n)). Note tht R(n) is function of n, where n is the numer of right endpoints (or the numer of rectngles). Copyright 1992-2 Roert E. Kowlczyk & Adm O. Husknecht, Deprtment of Mthemtics, UMss Drtmouth; Revised 8/11.
L 11: Approximte Integrtion 2 To gin some insight into the properties of the right endpoint rule, let's evlute R(n) for incresingly lrger vlues of n nd oserve how well the vlues of R(n) pproximte the definite integrl. We cn use TEMATH's Expression Clcultor to evlute R(n) y following these instructions: Select Accurcy from the Options menu. Enter 15 for the numer of significnt digits nd click the OK utton. The results of ll clcultions performed in the Expression Clcultor window will now e displyed with fifteen significnt digits. Select Clcultors Expression Clcultor from the Work menu. Enter R(1). Press the Enter key. Be sure tht the flshing cursor remins on the sme line s R(1). The right endpoint rule will e evluted using ten rectngles nd its vlue will e written on the next line in the Expression Clcultor window. An pproximtion is sid to hve d correct deciml digits of ccurcy if exct vlue pproximte vlue <.5 1 d. For our prticulr exmple, this ecomes 1 pproximte vlue <.5 1 d. 1. ) Wht is the vlue of the pproximtion R(1)?... ) Wht is the solute error sin(x)dx R(1) of the pproximtion?... To gin some insight into the reltionship etween the solute error nd the vlue of n, let's write the vlue of the solute error in the form c, where c is constnt nd n is the n numer of rectngles. For exmple, if the solute error is.76.8 nd n = 1, then we cn solve the eqution.8 = c /1 for c to get c =.8 nd write the solute error in the form sin(x)dx R(1).8. We will use this form of the solute error 1 in the following exmples. Next, let's chnge R(1) to R(1) nd press the Enter key. The right endpoint rule is now evluted using one hundred rectngles nd its vlue is written on the next line in the Expression Clcultor window. 2. ) Wht is the vlue of the pproximtion R(1)?... ) Wht is the solute error sin(x)dx R(1) of the pproximtion?... Kowlczyk & Husknecht 8/14/
L 11: Approximte Integrtion 3 d) Write the vlue of the solute error in the form c n... e) How mny more correct deciml digits re there in this pproximtion thn the previous one with n = 1?... Chnge R(1) to R(1) nd press the Enter key. The right endpoint rule is evluted using one thousnd rectngles nd its vlue is written on the next line. 3. ) Wht is the vlue of the pproximtion R(1)?... ) Wht is the solute error sin(x)dx R(1) of the pproximtion?... d) Write the vlue of the solute error in the form c n... e) How mny more correct deciml digits re there in this pproximtion thn the previous one with n = 1?... If there is constnt k such tht the solute error cn e written s exct vlue pproximte vlue < k n for ll n >, then we sy tht the pproximtions converge linerly to the exct vlue s n increses. 4. Bsed on the ove clcultions, guess vlue for k.... 5. If the pproximtions converge linerly to the exct vlue, how mny more correct deciml digits re there in the pproximtion when n is incresed y fctor of 1? Explortion 2 Trpezoidl Rule The vlue of the definite integrl rule f (x)dx cn e pproximted using the trpezoidl f (x)dx T n = x 2 n 1 f () + 2 f (x i ) + f () i=1 Kowlczyk & Husknecht 8/14/
L 11: Approximte Integrtion 4 where x = ( )/n nd x i = + i x. Let's use the trpezoidl rule to pproximte the definite integrl sin( x)dx =1 given in the previous explortion. If you've lredy done the previous explortion (entered,, nd f(x)), you only need to enter the trpezoidl rule y doing the following: Select New Function from the Work menu. Delete the defult function nme nd enter the trpezoidl rule (press the Option w key for ) T(n) = ( )/(2n) (f()+2 (i, 1, n 1, f(+i( )/n))+f()). Note tht T(n) is function of n, where n is the numer of trpezoids used in pproximting the integrl on the intervl [, ]. Use the Expression Clcultor (s descried in the previous explortion) to evlute T(1), T(1), nd T(1). 1. ) Wht is the vlue of the pproximtion T(1)?... ) Wht is the solute error sin(x)dx T(1) of the pproximtion?... d) Write the vlue of the solute error in the form c n... 2 2. ) Wht is the vlue of the pproximtion T(1)?... ) Wht is the solute error sin(x)dx T(1) of the pproximtion?... c) How mny correct deciml digits re there in the pproximtion?... d) Write the vlue of the solute error in the form c n... 2 e) How mny more correct deciml digits re there in this pproximtion thn the previous one with n = 1?... 3. ) Wht is the vlue of the pproximtion T(1)?... ) Wht is the solute error sin(x)dx T(1) of the pproximtion?... d) Write the vlue of the solute error in the form c n... 2 Kowlczyk & Husknecht 8/14/
L 11: Approximte Integrtion 5 e) How mny more correct deciml digits re there in this pproximtion thn the previous one with n = 1?... If there is constnt k such tht the solute error cn e written s exct vlue pproximte vlue < k n 2 for ll n >, then we sy tht the convergence of the pproximtions to the exct vlue is qudrtic. 4. Bsed on the ove clcultions, guess vlue for k.... 5. If the convergence is qudrtic, how mny more correct deciml digits re there in the pproximtion when n is incresed y fctor of 1?... Explortion 3 Simpson's Rule The vlue of the definite integrl f (x)dx cn e pproximted using Simpson's rule [ ] f (x)dx S n = x 3 f () + 4 f (x ) + 2 f (x ) + 4 f (x )+... +2 f (x ) + 4 f (x ) + f () 1 2 3 n 2 n 1 where x = ( )/n, x i = + i x, nd n is n even positive integer. Let's use Simpson's Rule to pproximte the sme definite integrl sin( x)dx =1 given in the previous explortions. If you've lredy done the previous explortion (entered,, nd f(x)), you only need to enter Simpson's rule y doing the following Select New Function from the Work menu. Delete the defult function nme nd enter Simpson's rule (press the Option w key for ) S(n) = ( )/(3n) (f()+2 (i,1,n 1,f(+2i( )/n))+ 4 (i,1,n,f(+(2i 1)( )/n))+f()). Use the expression clcultor (s descried in the previous explortion) to evlute S(1), S(1), nd S(1). 1. ) Wht is the vlue of the pproximtion S(1)?... ) Wht is the solute error sin(x)dx S(1) of the pproximtion?... Kowlczyk & Husknecht 8/14/
L 11: Approximte Integrtion 6 d) Write the vlue of the solute error in the form c n 4... 2. ) Wht is the vlue of the pproximtion S(1)?... ) Wht is the solute error sin(x)dx S(1) of the pproximtion?... d) Write the vlue of the solute error in the form c n... 4 e) How mny more correct deciml digits re there in this pproximtion thn the previous one with n = 1?... 3. ) Wht is the vlue of the pproximtion S(1)?... ) Wht is the solute error sin(x)dx S(1) of the pproximtion?... d) Write the vlue of the solute error in the form c n... 4 e) How mny more correct deciml digits re there in this pproximtion thn the previous one with n = 1?... If there is constnt k such tht the solute error cn e written s exct vlue pproximte vlue < k n p for ll n >, then we sy tht the convergence of the pproximtions to the exct vlue is of the order p. 4. Bsed on the ove clcultions, guess vlue for k.... 5. If the convergence is of the order 4, how mny more correct deciml digits re there in the pproximtion when n is incresed y fctor of 1?... 6. ) Which of the three pproximte integrtion methods is the most efficient, tht is, which method uses the fewest function evlutions to chieve prticulr ccurcy? Kowlczyk & Husknecht 8/14/
L 11: Approximte Integrtion 7. ) Explin your nswer to prt ).... c) Descrie the procedure you would use to pproximte the vlue of definite integrl (whose exct vlue you did not know) to ten correct deciml digits... Explortion 4 Error Bounds Are the ptterns of error reduction found in the previous explortions the sme for ny definite integrl? To investigte this question, let's pproximte nother known definite integrl 4 xdx = 5.3333K y using the methods descried in the first three explortions. First, let's set up TEMATH y doing the following: Set =, = 4, nd f(x) = sqrt(x) in the Work window. 1. Use the Expression Clcultor to find the following: ) R(1)... R(1)... R(1)... ) T(1)... T(1)... T(1)... c) S(1)... S(1)... S(1)... Kowlczyk & Husknecht 8/14/
L 11: Approximte Integrtion 8 2. As the vlue of n incresed from 1 to 1 to 1, ws the pttern of the reduction of the pproximtion error the sme s in the previous explortions?... Explin.... At this point, you should refer to the section in your clculus text ook on pproximte integrtion techniques nd red it crefully. It cn e shown tht if f (x) M R, f (x) M T, nd f (4) (x) M S for x (tht is, there exist set of constnts M R, M T, nd M S which ound the vlues of the derivtives of f (x) ), then the solute errors in the Right Endpoint Rule, the Trpezoidl Rule, nd Simpson's Rule re ounded s follows: E R M R ( )2 2n, E T M T ( )3, nd E 12n 2 S M S ( )5 18n 4 3. ) For f (x) = sin(x), find ound M R such tht f (x) M R for x / 2... ) Find the theoreticl vlue of k such tht E R M R ( )2 2n = k n... c) Are the results of the first explortion consistent with the theoreticl error ound clculted in prt )? Explin... Kowlczyk & Husknecht 8/14/
L 11: Approximte Integrtion 9 4. Use the theoreticl error ounds given ove to explin the ptterns of the pproximtion errors oserved in the first three explortions for the Right Endpoint Rule, the Trpezoidl Rule, nd Simpson's Rule.... 5. Use the theoreticl error ounds given ove to explin why the pttern of pproximtion error reduction chnged in explortion 4 with f (x) = x. Give specil ttention to the derivtives of f (x) on the intervl (, 4)... Kowlczyk & Husknecht 8/14/
L 11: Approximte Integrtion 1 Explortion 5 Evluting Integrls 1 The integrl sin(x 2 )dx hs no closed form solution. Use the Expression Clcultor to find n pproximtion to this integrl tht hs twelve correct deciml digits. 1 1. ) sin(x 2 )dx... ) Explin how you otined the nswer given in prt ) nd give resons why the pproximtion is correct to twelve deciml digits.... Kowlczyk & Husknecht 8/14/