Rose-Hulmn Undergrdute Mthemtics Journl Volume 16 Issue 2 Article 8 Tngent Line nd Tngent Plne Approximtions of Definite Integrl Meghn Peer Sginw Vlley Stte University Follow this nd dditionl works t: http://scholr.rose-hulmn.edu/rhumj Recommended Cittion Peer, Meghn (2015) "Tngent Line nd Tngent Plne Approximtions of Definite Integrl," Rose-Hulmn Undergrdute Mthemtics Journl: Vol. 16 : Iss. 2, Article 8. Avilble t: http://scholr.rose-hulmn.edu/rhumj/vol16/iss2/8
Rose- Hulmn Undergrdute Mthemtics Journl Tngent Line nd Tngent Plne Approximtions of Definite Integrls Meghn Peer Volume 16, No. 2, Fll 2015 Sponsored by Rose-Hulmn Institute of Technology Deprtment of Mthemtics Terre Hute, IN 47803 Emil: mthjournl@rose-hulmn.edu http://www.rose-hulmn.edu/mthjournl Sginw Vlley Stte University
Rose-Hulmn Undergrdute Mthemtics Journl Volume 16, No. 2, Fll 2015 Tngent Line nd Tngent Plne Approximtions of Definite Integrls Meghn Peer Abstrct. Oftentimes, it becomes necessry to find pproximte vlues for definite integrls, since the mjority cnnot be solved through direct computtion. The methods of tngent line nd tngent plne pproximtion cn be derived s methods of integrl pproximtion in two nd three-dimensionl spces, respectively. Formuls re derived for both methods, nd these formuls re compred with existing methods in terms of efficiency nd error. Acknowledgements: I would like to thnk Dr. Emmnuel Kengni Ncheuguim for his techings, guidnce, nd continued support. I lso grtefully cknowledge the Student Reserch nd Cretivity Institute t Sginw Vlley Stte University for their finncil support in this endevor.
Pge 138 RHIT Undergrd. Mth. J., Vol. 16, No. 2 1 Introduction to Definite Integrls nd Approximtion A definite integrl is defined s limit of Riemnn sums; therefore ny Riemnn sum could be used s n pproximtion to the integrl. Consider function f defined on closed intervl [, b. Divide the intervl [, b into n subintervls of equl width x = b. We let n x 0 (= ), x 1, x 2,..., x n (= b) be the endpoints of these subintervls. The definite integrl of f from to b (illustrted by Figure 1 below) is b f(x)dx = lim n n f(x i ) x i=1 provided tht the limit exists, where x i is ny point in the ith subintervl [x i 1, x i, i = 1, 2,..., n [3, p. 296. If the limit does exist, f is sid to be integrble on [, b. Figure 1: If f(x) 0, the integrl b f(x)dx is the re under the curve y = f(x) from to b. The left endpoint pproximtion, right endpoint pproximtion, nd Midpoint Rule ll represent pproximtions of the definite integrl by the use of rectngles. These three pproximtions re illustrted below for comprison purposes, where x = b, nd x n i is the midpoint of [x i 1, x i. The pproximtions L n, R n, nd M n re the left endpoint, right endpoint, nd midpoint pproximtions, respectively: b f(x)dx L n = n f(x i 1 ) x i=1 b f(x)dx R n = n f(x i ) x i=1
RHIT Undergrd. Mth. J., Vol. 16, No. 2 Pge 139 b f(x)dx M n = n f(x i ) x. (1.1) The Trpezoidl Rule pproximtion, T n, essentilly is result of verging the right endpoint nd left endpoint pproximtions. Figurtively, this method of pproximtion is derived from the ccumultion of res of trpezoids, rther thn rectngles, nd is given by b i=1 f(x)dx T n = x [ f(x 0 ) + 2f(x 1 ) + 2f(x 2 ) + + 2f(x n 1 ) + f(x n ). (1.2) 2 Another pproximtion technique, Simpson's Rule, pproximtes the definite integrl using prbols insted of stright line segments [3, p. 535. This pproximtion, S n is given by b f(x)dx S n = x [ f(x 0 ) + 4f(x 1 ) + 2f(x 2 ) + 4f(x 3 ) + 3 + 2f(x n 2 ) + 4f(x n 1 ) + f(x n ) where n is n even number. In this pper, we study n dditionl pproximtion technique. Specificlly, we will pproximte definite integrl of function by using the tngent line pproximtion to the function t the left endpoint of the subintervls. We introduce this technique in Section 2.1, below, nd compute bound on its error in Section 2.2. In Section 2.3 we compre this error to the error in the Midpoint Rule nd Trpezoidl Rule explicitly, nd experimentlly to the error in the left nd right endpoint methods. We provide n explicit exmple in Section 2.4. In Section 3, we extend the ides of Section 2 to consider pproximtions of definite integrls of functions of two vribles. As such, we consider pproximtions of the definite integrl by tngent plnes, compute n error bound, nd conclude with n exmple. Finlly, we provide n introduction to more expnsive pproch to the tngent line method, involving the pproximtion of curve using ny Tylor polynomil of degree n. 2 Tngent Line Approximtion of Definite Integrls nd Error Formul 2.1 Left Endpoint Tngent Line Approximtion In this section, we begin our study of using tngent lines in the clcultion of definite integrl pproximtions. Given differentible function f, we pproximte f on ech subintervl
Pge 140 RHIT Undergrd. Mth. J., Vol. 16, No. 2 [x i, x i+1 by computing the tngent line to f t the left endpoint x i. We then pproximte the definite integrl of f on the intervl [, b by summing the integrls of these tngent line pproximtions. We will denote the tngent line on [x i, x i+1 by T i (x), so tht T i (x) = f (x i )(x x i ) + f(x i ) nd xi+1 x i T i (x)dx = xi+1 x i [ f (x i )(x x i ) + f(x i ) dx. Since we wnt the ccumulted vlues of ech integrl for ech subintervl, we must sum ech integrl of the form bove. This gives n 1 xi+1 n 1 = n 1 = n 1 = x i [ f (x i )(x x i ) + f(x i ) dx [ ( f (xi+1 ) 2 (x i ) (x i) 2 ) x i (x i+1 x i ) + f(x i )(x i+1 x i ) 2 2 ) f (x i )( 2 (x i+1 + x i ) x (x i ) x + f(x i ) x ) x( 2 f (x i ) x + f(x i ). Therefore, the formul for the tngent line pproximtion of definite integrls (illustrted by Figure 2 below), using left endpoints, is given by b n 1 ) f(x)dx x( 2 f (x i ) x + f(x i ). (2.1) Remrk: The limittion to using this formul in clcultions resides in the fct tht in order to use (2.1) to pproximte n integrl, it is necessry tht the given function is differentible, since we must be ble to compute the derivtive. Given this limittion, we cn derive similr formul by pproximting the derivtive, in which cse, we hve [3, p. 114 f (x i ) f(x i + x) f(x i ). (2.2) x
RHIT Undergrd. Mth. J., Vol. 16, No. 2 Pge 141 Figure 2: Tngent line pproximtion using left endpoints. Substituting this into (2.1) gives b n 1 f(x)dx n 1 = n 1 = [ ( 1 x 2 [ f(xi + x) f(x i ) x ) x( 2 (f(x i) + f(x i + x) ) x( 2 (f(x i) + f(x i+1 ). ) x + f(x i ) This formul is simply the verge of the right endpoint pproximtion nd the left endpoint pproximtion, which is the Trpezoidl Rule given in (1.2). Therefore, the tngent line pproximtion is equivlent to the Trpezoidl Rule when the derivtive is pproximted by (2.2). 2.2 Error Formul for Tngent Line Approximtion In this section, we compute bound on the error in the tngent line pproximtion. Lter, we will compre this error formul to the error bounds in the Trpezoidl nd Midpoint rules. Note tht Simpson's Rule provides the lest error for smooth functions nd is not stright line pproximtion like the other methods described bove. As such, we will not refer to Simpson's Rule in the following comprisons nd exmples. The error bound for the tngent line pproximtion, E T, is given in the following theorem.
Pge 142 RHIT Undergrd. Mth. J., Vol. 16, No. 2 Theorem 2.1. Let [, b be the intervl of integrtion nd n be the number of subintervls within the intervl [, b. Assume f is continuous over [, b, nd let ζ 0 [, b be such tht f(ζ 0 ) = mx ζ b f(ζ). Then the error, E T, for the tngent line pproximtion stisfies E T (b )3 6n 2 f (ζ 0 ). The reminder of this section will prove Theorem 2.1. Assume the function f is twice differentible on (, b). To find the totl error for the tngent line pproximtion, the sum of the errors for ech intervl must be clculted [2. Thus we hve the totl error, E T, given by where E i is the error for the ith subintervl. n 1 E T = E i, Initilly, we found the formul for the pproximtion by integrting ech tngent line of the form T i (x) = f (x i )(x x i ) + f(x i ) (where x i is the left endpoint of the ith subintervl) nd summing over ech intervl. Within this clcultion, however, lies degree of inccurcy, especilly when n (the number of subintervls) is reltively smll. To quntify this inccurcy, let s write f(x) = T i (x) + R i (x) so tht R i (x) is the reminder in the first-degree Tylor pproximtion of f. Then R i (x) stisfies R i (x) = f (ζ i ) (x x i ) 2 2! for some ζ i [x i, x (where ζ i depends on x) [1, p. 3. Assume f is continuous on n intervl [, b. Then by the Intermedite Vlue Theorem [1, p. 7, there exists ζ 0 [, b such tht f (ζ 0 ) = mx ζ b f (ζ) f (ζ i ), i = 0, 1, 2,..., n 1. Therefore R i (x) f (ζ 0 ) (x x i ) 2, i = 0, 1, 2,..., n 1. 2!
RHIT Undergrd. Mth. J., Vol. 16, No. 2 Pge 143 Thus, we find the error, E i, over single intervl [x i, x i+1 by integrting the bove inequlity over the subintervl: xi+1 ( E i = f(x) Ti (x) ) xi+1 dx f(x) T i (x) dx x i = x i xi+1 x i f (ζ 0 ) 2! Thus, the error for one subintervl [x i, x i+1 is bounded by R i (x) dx xi+1 x i (x x i ) 2 dx = f (ζ 0 ) (x i+1 x i ) 3. 6 where x = x i+1 x i. E i x3 6 f (ζ 0 ), The totl error is simply the sum of the errors from ech segment. And so the totl error in the tngent pproximtion is n 1 n 1 x 3 E T = E i 6 f (b )3 (ζ 0 ) f (ζ 6n 2 0 ). 2.3 Comprison with Other Methods The error given in Theorem 2.1 decreses s n gets lrger, nd the pproximtion becomes more exct s n. When choosing the best method to pproximte the definite integrl, few fctors must be tken into considertion. In order to pply the tngent line pproximtion, the function must be differentible, since we must tke the derivtive in the computtion. We sw erlier tht simply substituting the pproximtion of the derivtive into the derivtive itself gives us the pproximtion for the Trpezoidl Rule. The error for the Trpezoidl Rule is given in similr form s tht in Theorem 2.1: E T r (b )3 12n 2 f (ζ 0 ) [2. (2.3)
Pge 144 RHIT Undergrd. Mth. J., Vol. 16, No. 2 Notice the only difference between the two equtions is in the denomintor. Since the denomintor in (2.3) is twice s lrge, the Trpezoidl Rule in generl pproximtes the definite integrl more efficiently thn the tngent line pproximtion. With this reliztion in mind, it would be worthwhile to sk if we could improve the ccurcy of the tngent line pproximtion. For exmple, wht if insted of clculting the tngent line t the left endpoint of ech subintervl we clculted the tngent line t the midpoint? This pproximtion using the midpoint is over the sme intervl [, b with the subintervls = x 0, x 1, x 2,..., x n = b. Here we tke the midpoint, x i, of ech subintervl [x i, x i+1, nd derive formul in similr wy s (2.1), simply substituting the left endpoint x i with x i in the originl eqution of our tngent line over ech subintervl. Thus, we hve n 1 xi+1 n 1 = x i [ f (x i )(x x i ) + f(x i ) dx [ ( 1 ) f (x i ) 2 (x2 i+1 x 2 i ) (x i ) x + f(x i ) x. Substituting x i = 1 2 (x i+1 + x i ) x = x i+1 x i nd gives x 2 i+1 x 2 i = (x i+1 + x i )(x i+1 x i ) n 1 ( 1 ) ) f (x i )( 2 (x i+1 + x i )(x i+1 x i ) 2 (x i+1 + x i ) (x i+1 x i ) + f(x i ) x n 1 = f(x i ) x. This is equivlent to the Midpoint Rule given in (1.1). Note from geometry tht given vlue f( x i ), the re under ny stright line pssing through f( x i ) over tht subintervl will be the sme. This is illustrted in Figure 3. The Midpoint Rule is the prticulr cse when the line is horizontl. Therefore, ltering the tngent line pproximtion to use the midpoints of the subintervls insted of the left endpoint does not yield new pproximtion. However, it does improve ccurcy, since the error formul for the Midpoint Rule is given by E M (b )3 24n 2 f (ζ) [3, p. 534.
RHIT Undergrd. Mth. J., Vol. 16, No. 2 Pge 145 Figure 3: Given ny two lines pssing through the midpoint ( x i, f( x i )) of subintervl, the re beneth ech line will be the sme. Note here tht the shded re bove nd below the horizontl line is the sme for ech subintervl. The difference between the errors for the Midpoint Rule nd the tngent line pproximtion is, once gin, the denomintor. The denomintor for the Midpoint Rule is four times lrger thn tht for the tngent line pproximtion, indicting tht, in generl, the Midpoint Rule will give more ccurte pproximtion. Although the tngent line pproximtion seems to be less efficient compred with the Midpoint nd Trpezoidl Rules, repeted exmples hve illustrted tht this new pproximtion performs better thn the left nd right endpoint pproximtions. This is resonble conclusion, since both the left nd right endpoint pproximtions tend to significntly under-pproximte nd over-pproximte the re underneth the curve for positive function compred with the other methods mentioned. However, the ccurcy nd dependence of ech pproximtion depends on the given function to which we re pplying our pproximtion. 2.4 Applying the Tngent Line Approximtion Let s mnully pply the tngent line pproximtion to pproximte the re beneth the curve f(x) = sin 2 (x), from x = 1 to x = 5 with n = 4 subintervls. Here, we hve x = 5 1 4 = 1, nd f (x) = 2 sin(x) cos(x).
Pge 146 RHIT Undergrd. Mth. J., Vol. 16, No. 2 Applying the formul derived in (2.1), we obtin 5 1 f(x)dx = 3 ) x( 2 f (x i ) x + f(x i ) 3 2 f (x i ) + f(x i ) = (1) 2 (2 sin(1) cos(1)) + sin2 + 2 (2 sin(2) cos(2)) + sin2 (2) + 2 (2 sin(3) cos(3)) + sin2 (3) + 2 (2 sin(4) cos(4)) + sin2 (4) 2.558778942 rounding to nine deciml plces. We cn compre this pproximtion to the exct vlue of the definite integrl: 5 rounding to 9 deciml plces. 1 sin 2 (x)dx = 1 [x 1 2 2 sin(2x) 5 = 2.363329634 1 Note tht we chose n to be reltively smll in order to simplify the computtionl demonstrtion of the method. As n grows lrger, the ccurcy of the pproximtion increses substntilly. Below, we show the result of the tngent line pproximtion compred with the Midpoint Rule, Trpezoidl Rule, nd Left Endpoint Approximtion, s n increses. n-subintervls Tngent Line Midpoint Trpezoidl Left Endpoint n = 10 2.3995948697 2.3732023841 2.3437421589 2.3014496896 n = 50 2.3648614575 2.3637174756 2.3625542003 2.3540957064 n = 100 2.3637149095 2.3634265404 2.3631358380 2.3589065910 n = 500 2.3633451184 2.3633335100 2.3633218834 2.3624760340 The bove clcultions were quickly executed using the mthemticl softwre Mtlb. Note s n increses, ech pproximtion increses in ccurcy. In prticulr, the tngent line pproximtion rpidly decreses in error s the number of subintervls increses. This rpid decrese in error is illustrted in Figure 4 tht follows.
RHIT Undergrd. Mth. J., Vol. 16, No. 2 Pge 147 Figure 4: Grph of the pproximte error for the tngent line pproximtion s the number of subintervls n increses. 3 Tngent Plne Approximtion of Definite Integrls nd Error Formul 3.1 Introduction to Volumes nd Double Integrls We will now extend the ides of Section 2 to integrls of functions of two vribles. We consider function f of two vribles defined on closed region R = [, b [c, d with (x, y) in the two-dimensionl spce of rel numbers such tht x b nd c y d. The grph of f is surfce with eqution z = f(x, y) (illustrted by Figure 5 below). The region R is the projection of the surfce onto the xy plne. We wish to find the volume of the surfce S which lies bove R nd below the grph of the positive function f [3, p. 998. Figure 5: Three-dimensionl grph of surfce f(x, y).
Pge 148 RHIT Undergrd. Mth. J., Vol. 16, No. 2 To find this volume, we divide the rectngle R into subrectngles. We divide the intervl [, b into m subrectngles [x i, x i+1 of equl width x = b nd divide [c, d into n subrectngles [y j, y j+1 of equl width y = d c. m n And so, we find the volume beneth f by double integrting over the region R s follows, for smple points (x ij, y ij) in [x i, x i+1 [y i, y i+1 nd A = x y, R f(x, y)da = lim n,m m i=1 n f(x ij, yij) A [3, p. 1000. j=1 3.2 Tngent Plne Approximtion The tngent plne method of pproximtion uses tngent plnes to pproximte the given function f of two vribles over ech subrectngle by choosing the lower left point of ech rectngle nd finding the liner pproximtion/tngent plne of f t tht point. This plne is given by T ij (x, y) = f x (x i, y j )(x x i ) + f y (x i, y j )(y y j ) + f(x i, y j ) [3, p. 941. Here, we let x i = x 0 + i x nd y j = y 0 + j y, where i = 1, 2,..., m nd j = 1, 2,..., n. The vlue of the integrl for one subrectngle is given by yj+1 xi+1 y j x i T ij (x, y)dxdy. Therefore, in order to find the combined vlue over ll the subrectngles, we must sum over the region R. This gives us our tngent plne pproximtion of definite integrls: m 1 n 1 yj+1 xi+1 j=0 y j x i [ f x (x i, y j )(x x i ) + f y (x i, y j )(y y j ) + f(x i, y j ) dxdy = m 1 n 1 j=0 [ [( 1 f x (x i, y j ) 2 (x i + x) 2 (x i )(x i + x) + 1 ) 2 (x i) 2 y [( 1 + f y (x i, y j ) 2 (y j + y) 2 (y j )(y j + y) + 1 ) 2 (y j) 2 x + f(x i, y j ) x y. Substituting x i+1 x i = x, y j+1 y j = y, x i+1 = x i + x, nd y j+1 = y j + y, we obtin
RHIT Undergrd. Mth. J., Vol. 16, No. 2 Pge 149 b d c f(x, y)dydx m 1 n 1 j=0 [ x y f x (x i, y j ) 1 2 x + f y(x i, y j ) 1 2 y + f(x i, y j ). (3.1) d Figure 6: Approximtion of the integrl b f(x, y)dydx using the tngent plne pproximtion, nd the volume of solids below tngent plnes t ech c subrectngle. Remember tht the tngent plne pproximtion bove uses tngent plnes found t the lower left corner of ech subrectngle. If, insted, we choose to find the tngent plne t the midpoint of ech subrectngle, we obtin the following fmilir Midpoint Rule for double integrls: m n f(x, y) f( x i, ȳ j ) A. R i=1 Note from geometry tht given the vlue f( x i, ȳ j ) t the midpoint, the volume under ny plne pssing through f( x i, ȳ j ) over the subrectngle [x i, x i+1 [y j, y j+1 will be the sme. The Midpoint Rule is the prticulr cse when the plne is horizontl nd will be equivlent to the tngent plne pproximtion t the midpoint. Remrk: Similr to the tngent line pproximtion, the limittion to using the tngent plne pproximtion in clcultions resides in the fct tht in order to use (3.1) to pproximte definite integrls, it is necessry tht the given function be differentible. Tking this limittion into considertion, we cn derive similr formul to the tngent plne pproximtion by using the pproximtions of the prtil derivtives. Thus, we hve the following substitutions: j=1 f x (x i, y j ) f(x i + x, y j ) f(x i, y j ) x
Pge 150 RHIT Undergrd. Mth. J., Vol. 16, No. 2 nd f y (x i, y j ) f(x i, y j + y) f(x i, y j ) y [3, p. 926. Substituting these equtions into (3.1) gives b d c m 1 n 1 [( f(xi + x, y j ) f(x i, y j ) ) 1 f(x, y)dydx x y x 2 x j=0 ( f(xi, y j + y) f(x i, y j ) ) 1 + y 2 y + f(x i, y j ) m 1 n 1 = x y 1 [ f(x i+1, y j ) + f(x i, y j+1 ). 2 j=0 Note tht when we pproximte the derivtive, s bove, we obtin the three-dimensionl Trpezoidl Rule. We re tking the height for prticulr subrectngle to be the verge of the heights t two corners (similr to the two-dimensionl Trpezoidl Rule which tkes the height to be the verge of the two endpoints). 3.3 Error Formul for Tngent Plne Approximtion The error formul for the tngent plne pproximtion cn be derived using similr methods tht were used to derive the error formul for the tngent line pproximtion. We cn find the totl error for the three-dimensionl tngent plne pproximtion by tking the sum of the errors for ech subrectngle. Therefore, the totl error of the pproximtion, E T P, is given by n m E T P = i=1 where E ij is the error for the (i, j)th subrectngle. The formul for the tngent plne pproximtion ws initilly found by double integrting the formul of the tngent plne nd summing over ll the subrectngles. As with the tngent line pproximtion, the tngent plne pproximtion contins degree of inccurcy, especilly when the number of subrectngles (n nd m) is reltively smll. The error bound for the tngent plne pproximtion, E T P, is given in the following theorem. j=1 E ij
RHIT Undergrd. Mth. J., Vol. 16, No. 2 Pge 151 Theorem 3.1. Let [, b [c, d be the region of integrtion. We divide the intervl [, b into m subrectngles of equl width nd divide [c, d into n subrectngles of equl width. Suppose tht M is n upper bound on f xx, f xy, nd f yy over the region [, b [c, d, nd f xx, f xy, nd f yy re continuous on [, b [c, d. Then the error for the tngent plne pproximtion is given by [ (b ) 3 (d c) E T P M + (b )2 (d c) 2 (b )(d c)3 +. 6n 2 4nm 6m 2 The reminder of this section will prove Theorem 3.1. We let T ij (x, y) = f x (x i, y j )(x x i ) + f y (x i, y j )(y y j ) + f(x i, y j ) be the first degree Tylor polynomil in two vribles of f, nd R ij the reminder (error) of the Tylor series on the subrectngle [x i, x i+1 [y i, y i+1 such tht f(x, y) = T ij (x, y) + R ij (x, y) [1, p. 4. Assume f xx, f xy, nd f yy re continuous over the region R = [, b [c, d, nd suppose tht M is n upper bound on f xx, f xy, nd f yy on R. Then the reminder R ij stisfies the inequlity R ij (x, y) M 2! [(x x i ) 2 + 2 (x x i )(y y j ) + (y y j ) 2. (3.2) We clculte the error over single subrectngle [x i, x i+1 [y i, y i+1 by double integrting the error R ij over the subrectngle. And so, we hve tht the error for ech subrectngle is E ij = yj+1 xi+1 y j x i yj+1 xi+1 R ij (x, y)dxdy M [ (x xi ) 2 + 2(x x i )(y y j ) + (y y j ) 2 dxdy 2! y j x i = M 2 3 (x i+1 x i ) 3 (y j+1 y j ) + 1 2 (x i+1 x i ) 2 (y j+1 y j ) 2 + 1 3 (x i+1 x i )(y j+1 y j ) 3 = M 6 ( x)3 ( y) + 1 4 ( x)2 ( y) 2 + 1 6 ( x)( y)3 where x i+1 x i = x nd y j+1 y j = y. Note tht the bsolute vlue on the right hnd side of (3.2) is removed during integrtion, since we re only considering vlues of x nd y tht re greter thn x i nd y j, respectively.
Pge 152 RHIT Undergrd. Mth. J., Vol. 16, No. 2 We find the totl error by summing over ll subrectngles. So, the totl error for the tngent plne pproximtion is where x = b n E T P n i=1 m j=1 M 6 ( x)3 ( y) + 1 4 ( x)2 ( y) 2 + 1 6 ( x)( y)3 [ nm M 6 ( x)3 ( y) + nm 4 ( x)2 ( y) 2 + nm 6 ( x)( y)3 [ (b ) 3 (d c) M + (b )2 (d c) 2 (b )(d c)3 + 6n 2 4nm 6m 2 d c nd y =. m 3.4 Applying the Tngent Plne Approximtion Let s mnully pply the tngent plne pproximtion to pproximte the volume beneth the surfce f(x, y) = x 3y 2, where 0 x 2 nd 1 y 2. For simplified ppliction of the method, ssign subrectngles m nd n such tht m = n = 2. We hve tht x = b m = 2 0 = 1 nd y = d c 2 n = 2 1 2 = 1 2. The prtil derivtives necessry for the computtion re f x (x, y) = 1 nd f y (x, y) = 6y. Applying the formul derived in (3.1), we obtin 2 2 0 1 f(x, y)dydx 1 1 j=0 [ x y f x (x i, y j ) 1 2 x + f y(x i, y j ) 1 2 y + f(x i, y j ) ( 1 ( 1 ( 1 )( 1 ( 1 ( = (1) f x (0, 1) (1) + f y (0, 1) + f(0, 1) + (1) x 0, 2)[ 2) 2 2) 2)[f 3 )( 1 (1) 2 2) ( + f y 0, 3 )( 1 )( 1 ( + f 0, 2 2 2) 3 ) ( 1 ( 1 ( 1 )( 1 + (1) f x (1, 1) (1) + f y (1, 1) 2 2)[ 2) 2 2) ( 1 ( + f(1, 1) + (1) x 1, 2)[f 3 )( 1 ( + f y 1, 2 2)(1) 3 )( 1 )( 1 ( + f 1, 2 2 2) 3 ) 2 = 11.50. We hve tht 2 0 2 1 (x 3y 2 )dydx = 12. Therefore, the error in our clcultion is 12 ( 11.50) = 0.50. We chose m nd n to be very smll in order to simplify the computtionl demonstrtion of the method. As the number of subrectngles m nd n grow lrge, the error decreses
RHIT Undergrd. Mth. J., Vol. 16, No. 2 Pge 153 substntilly. Below, we show the result of the tngent plne pproximtion compred with the Midpoint Rule nd Trpezoidl Rule (both three-dimensionl), s m nd n increse substntilly. nm-subrectngles Tngent Plne Midpoint Trpezoidl m = n = 10 11.980000000 11.994999999 12.010000000 m = n = 50 11.999200000 11.999800000 12.000400000 m = n = 100 11.999800000 11.999950000 12.000100000 The bove clcultions were quickly executed using the mthemticl softwre Mtlb. Note s m nd n increse, ech pproximtion increses in ccurcy. In prticulr, the tngent plne pproximtion rpidly decreses in error s the number of subrectngles is mximized. This rpid decrese in error is illustrted in Figure 7 tht follows. Figure 7: Grph of the pproximte error for the tngent plne pproximtion s the number of subrectngles m nd n increses. 4 Conclusion In the preceding sections, we derived two lternte methods for pproximting definite integrls in two nd three-dimensionl spces, respectively. The tngent line pproximtion involves clculting the tngent lines t the left endpoints of subintervls over the region of integrtion. For positive function, the formul is essentilly the result of summing the res beneth ech tngent line corresponding to ech subintervl. In similr fshion, the tngent plne pproximtion involves clculting the tngent plnes t the lower left corners of subrectngles over the rectngulr region of integrtion. Figurtively, for positive function of two vribles, the formul is the result of summing the volumes beneth ech tngent plne corresponding to ech subrectngle.
Pge 154 RHIT Undergrd. Mth. J., Vol. 16, No. 2 5 Future Work Note tht both the tngent line nd tngent plne methods cn be extended to other Tylor pproximtions in hopes of decresing the error nd incresing the ccurcy. For the tngent line pproximtion, we pproximted the function f using the eqution of the tngent line, which is the first degree Tylor polynomil of f. We then used this eqution to derive formul for pproximting definite integrls. We cn extend this ide to pproximting the function f using polynomils of lrger degrees insted of simply using tngent lines. Therefore, we could pproximte the given curve using Tylor polynomils of f, nd then follow similr procedure for deriving definite integrl pproximtion. For instnce, if we wnt to pproximte f using polynomil of degree n, we would use the nth degree Tylor polynomil of f, given tht f is t lest n times differentible. References [1 J. H. HEINBOCKEL, Numericl Methods for Scientific Computing, Trfford Publishing, Victori, BC, Cnd, 2009. [2 A. KAW, M. KETELTAS, Trpezoidl Rule of Integrtion, Holistic Numericl Methods, Tmp, FL, 2012. [3 J. STEWART, Clculus: 7th ed., Brooks/Cole Cengge Lerning, Belmont, CA, 2011.