> On Dec 16, 2015, at 9:00 AM, Steven Stoline <sstoline at gmail.com> wrote: > > Dear William: *Left and Right Riemann Sums* > > > Is there is a way to modify your function to compute Left Riemann Sum and > Right Riemann Sum. I tried to modify yours, but i was not be able to make > it work correctly. > > This is your function used to compute the Middle Riemann Sum.I think it's actually Dalgaard's method.> > showIntegral.med <- function (f, xmin, xmax, n = 16) > { > curve(f(x), from = xmin, to = xmax, lwd = 2, col = "blue") > abline(h = 0) > dx <- (xmax - xmin)/n > right <- xmin + (1:n) * dx > left <- right - dx > mid <- right - dx/2 > fm <- f(mid) > rect(left, 0, right, fm, density = 20, border = "red") > points(mid, fm, col = "red", cex = 1.25, pch = 19) > sum(fm * dx) > } > > > > ### Example 1: f(x) = x^2 , xmin=-4, xmax=4 > ### ==============================> > > > showIntegral.med(f=function(x)x^2, xmin=-4, xmax=4, n=16)Wouldn't it just involve skipping the 'mid' calculations and using either the right or left values? Illustration for right: showIntegral.rt <- function (f, xmin, xmax, n = 16) { curve(f(x), from = xmin, to = xmax, lwd = 2, col = "blue") abline(h = 0) dx <- (xmax - xmin)/n right <- xmin + (1:n) * dx left <- right - dx fr <- f(right) rect(left, 0, right, fr, density = 20, border = "red") points(right, fr, col = "red", cex = 1.25, pch = 19) sum(fr * dx) } You can make it prettier with plotmath: showIntegral.rt <- function (f, xmin, xmax, n = 16) { curve(f(x), from = xmin, to = xmax, lwd = 2, col = "blue") abline(h = 0) dx <- (xmax - xmin)/n right <- xmin + (1:n) * dx left <- right - dx fr <- f(right) rect(left, 0, right, fr, density = 20, border = "red") points(right, fr, col = "red", cex = 1.25, pch = 19) sum(fr * dx) text(0,10, # might want to do some adaptive positioning instead bquote( integral( .(body(f) )*dx, a, b) == .( sum(fr * dx )) ) ) } -- David.> > > > with many thanks > steve > > On Sat, Nov 28, 2015 at 1:11 PM, William Dunlap <wdunlap at tibco.com> wrote: > >> Your right <- (1:n)*dx mean that your leftmost rectangle's left edge >> is at 0, but you want it to be at -4. You should turn this into a function >> so you don't have to remember how the variables in your code depend >> on one another. E.g., >> >> showIntegral <- function (f, xmin, xmax, n = 16) >> { >> curve(f(x), from = xmin, to = xmax, lwd = 2, col = "blue") >> abline(h = 0) >> dx <- (xmax - xmin)/n >> right <- xmin + (1:n) * dx >> left <- right - dx >> mid <- right - dx/2 >> fm <- f(mid) >> rect(left, 0, right, fm, density = 20, border = "red") >> points(mid, fm, col = "red", cex = 1.25, pch = 19) >> sum(fm * dx) >> } >>> showIntegral(f=function(x)x^2, xmin=-4, xmax=4, n=16) >> [1] 42.5 >>> showIntegral(f=function(x)x^2, xmin=-4, xmax=4, n=256) >> [1] 42.66602 >>> showIntegral(f=function(x)x^2, xmin=-4, xmax=4, n=1024) >> [1] 42.66663 >> >>> 2*4^3/3 >> [1] 42.66667 >>> showIntegral >> Bill Dunlap >> TIBCO Software >> wdunlap tibco.com >> >> >> On Fri, Nov 27, 2015 at 9:50 PM, Steven Stoline <sstoline at gmail.com> >> wrote: >>> Dear Peter: in my previous email I forgot to reply to the list too >>> >>> I used your code for more than one examples, and it works nicely. But >> when >>> I tried to use for the the function: f(x) = x^2, it looks like I am >> missing >>> something, but I could not figured it out. >>> >>> This what I used: >>> >>> >>> >>> f <- function(x) x^2 >>> >>> curve(f(x), from=-4, to=4, lwd=2, col="blue") >>> abline(h=0) >>> n <- 16 >>> dx <- 8/n >>> right <- (1:n)*dx >>> left <- right - dx >>> mid <- right - dx/2 >>> fm <- f(mid) >>> rect(left,0,right,fm, density = 20, border = "red") >>> points(mid, fm, col = "red", cex = 1.25, pch=19) >>> sum(fm*dx) >>> >>> >>> >>> 1/3 * (64+64) >>> >>> >>> >>> with many thanks >>> steve >>> >>> On Fri, Nov 27, 2015 at 3:36 PM, Steven Stoline <sstoline at gmail.com> >> wrote: >>> >>>> many thanks >>>> >>>> steve >>>> >>>> On Fri, Nov 27, 2015 at 9:20 AM, peter dalgaard <pdalgd at gmail.com> >> wrote: >>>> >>>>> Something like this? >>>>> >>>>> f <- function(x) x^3-2*x >>>>> curve(f(x), from=0, to=4) >>>>> abline(h=0) >>>>> n <- 16 >>>>> dx <- 4/n >>>>> right <- (1:n)*dx >>>>> left <- right - dx >>>>> mid <- right - dx/2 >>>>> fm <- f(mid) >>>>> points(mid, fm) >>>>> rect(left,0,right,fm) >>>>> >>>>> sum(fm*dx) >>>>> >>>>> 1/4 * 4^4 - 4^2 >>>>> >>>>> >>>>> -pd >>>>> >>>>> >>>>> On 27 Nov 2015, at 13:52 , Steven Stoline <sstoline at gmail.com> wrote: >>>>> >>>>>> Dear All: >>>>>> >>>>>> I am trying to explain to my students how to calculate the definite >>>>>> integral using the Riemann sum. Can someone help me to graph the area >>>>> under >>>>>> the curve of the function, showing the curve as well as the >> rectangles >>>>>> between 0 and 4.. >>>>>> >>>>>> *f(x) = x^3 - 2*x * >>>>>> >>>>>> over the interval [0 , 4] >>>>>> >>>>>> >>>>>> >>>>>> with many thanks >>>>>> steve >>>>>> >>>>>> -- >>>>>> Steven M. Stoline >>>>>> 1123 Forest Avenue >>>>>> Portland, ME 04112 >>>>>> sstoline at gmail.com >>>>>> >>>>>> [[alternative HTML version deleted]] >>>>>> >>>>>> ______________________________________________ >>>>>> R-help at r-project.org mailing list -- To UNSUBSCRIBE and more, see >>>>>> https://stat.ethz.ch/mailman/listinfo/r-help >>>>>> PLEASE do read the posting guide >>>>> http://www.R-project.org/posting-guide.html >>>>>> and provide commented, minimal, self-contained, reproducible code. >>>>> >>>>> -- >>>>> Peter Dalgaard, Professor, >>>>> Center for Statistics, Copenhagen Business School >>>>> Solbjerg Plads 3, 2000 Frederiksberg, Denmark >>>>> Phone: (+45)38153501 >>>>> Office: A 4.23 >>>>> Email: pd.mes at cbs.dk Priv: PDalgd at gmail.com >>>>> >>>>> >>>>> >>>>> >>>>> >>>>> >>>>> >>>>> >>>>> >>>>> >>>> >>>> >>>> -- >>>> Steven M. Stoline >>>> 1123 Forest Avenue >>>> Portland, ME 04112 >>>> sstoline at gmail.com >>>> >>> >>> >>> >>> -- >>> Steven M. Stoline >>> 1123 Forest Avenue >>> Portland, ME 04112 >>> sstoline at gmail.com >>> >>> [[alternative HTML version deleted]] >>> >>> ______________________________________________ >>> R-help at r-project.org mailing list -- To UNSUBSCRIBE and more, see >>> https://stat.ethz.ch/mailman/listinfo/r-help >>> PLEASE do read the posting guide >> http://www.R-project.org/posting-guide.html >>> and provide commented, minimal, self-contained, reproducible code. >> > > > > -- > Steven M. Stoline > 1123 Forest Avenue > Portland, ME 04112 > sstoline at gmail.com > > [[alternative HTML version deleted]] > > ______________________________________________ > R-help at r-project.org mailing list -- To UNSUBSCRIBE and more, see > https://stat.ethz.ch/mailman/listinfo/r-help > PLEASE do read the posting guide http://www.R-project.org/posting-guide.html > and provide commented, minimal, self-contained, reproducible code.David Winsemius Alameda, CA, USA
Dear David: could you please try it for the functions *f(x)=x^2* from *-4* to *4* and the function *f(x) = sqrt(x)* from *0* to *4*, and look watch the graphs. Thank you very much for your helps. steve On Wed, Dec 16, 2015 at 2:09 PM, David Winsemius <dwinsemius at comcast.net> wrote:> > > On Dec 16, 2015, at 9:00 AM, Steven Stoline <sstoline at gmail.com> wrote: > > > > Dear William: *Left and Right Riemann Sums* > > > > > > Is there is a way to modify your function to compute Left Riemann Sum and > > Right Riemann Sum. I tried to modify yours, but i was not be able to make > > it work correctly. > > > > This is your function used to compute the Middle Riemann Sum. > > I think it's actually Dalgaard's method. > > > > showIntegral.med <- function (f, xmin, xmax, n = 16) > > { > > curve(f(x), from = xmin, to = xmax, lwd = 2, col = "blue") > > abline(h = 0) > > dx <- (xmax - xmin)/n > > right <- xmin + (1:n) * dx > > left <- right - dx > > mid <- right - dx/2 > > fm <- f(mid) > > rect(left, 0, right, fm, density = 20, border = "red") > > points(mid, fm, col = "red", cex = 1.25, pch = 19) > > sum(fm * dx) > > } > > > > > > > > ### Example 1: f(x) = x^2 , xmin=-4, xmax=4 > > ### ==============================> > > > > > > > showIntegral.med(f=function(x)x^2, xmin=-4, xmax=4, n=16) > > Wouldn't it just involve skipping the 'mid' calculations and using either > the right or left values? Illustration for right: > > showIntegral.rt <- function (f, xmin, xmax, n = 16) > { > curve(f(x), from = xmin, to = xmax, lwd = 2, col = "blue") > abline(h = 0) > dx <- (xmax - xmin)/n > right <- xmin + (1:n) * dx > left <- right - dx > fr <- f(right) > rect(left, 0, right, fr, density = 20, border = "red") > points(right, fr, col = "red", cex = 1.25, pch = 19) > sum(fr * dx) > } > > You can make it prettier with plotmath: > > showIntegral.rt <- function (f, xmin, xmax, n = 16) > { > curve(f(x), from = xmin, to = xmax, lwd = 2, col = "blue") > abline(h = 0) > dx <- (xmax - xmin)/n > right <- xmin + (1:n) * dx > left <- right - dx > fr <- f(right) > rect(left, 0, right, fr, density = 20, border = "red") > points(right, fr, col = "red", cex = 1.25, pch = 19) > sum(fr * dx) > text(0,10, # might want to do some adaptive positioning instead > bquote( integral( .(body(f) )*dx, a, b) == .( sum(fr * dx )) ) ) > } > > -- > David. > > > > > > > > > with many thanks > > steve > > > > On Sat, Nov 28, 2015 at 1:11 PM, William Dunlap <wdunlap at tibco.com> > wrote: > > > >> Your right <- (1:n)*dx mean that your leftmost rectangle's left edge > >> is at 0, but you want it to be at -4. You should turn this into a > function > >> so you don't have to remember how the variables in your code depend > >> on one another. E.g., > >> > >> showIntegral <- function (f, xmin, xmax, n = 16) > >> { > >> curve(f(x), from = xmin, to = xmax, lwd = 2, col = "blue") > >> abline(h = 0) > >> dx <- (xmax - xmin)/n > >> right <- xmin + (1:n) * dx > >> left <- right - dx > >> mid <- right - dx/2 > >> fm <- f(mid) > >> rect(left, 0, right, fm, density = 20, border = "red") > >> points(mid, fm, col = "red", cex = 1.25, pch = 19) > >> sum(fm * dx) > >> } > >>> showIntegral(f=function(x)x^2, xmin=-4, xmax=4, n=16) > >> [1] 42.5 > >>> showIntegral(f=function(x)x^2, xmin=-4, xmax=4, n=256) > >> [1] 42.66602 > >>> showIntegral(f=function(x)x^2, xmin=-4, xmax=4, n=1024) > >> [1] 42.66663 > >> > >>> 2*4^3/3 > >> [1] 42.66667 > >>> showIntegral > >> Bill Dunlap > >> TIBCO Software > >> wdunlap tibco.com > >> > >> > >> On Fri, Nov 27, 2015 at 9:50 PM, Steven Stoline <sstoline at gmail.com> > >> wrote: > >>> Dear Peter: in my previous email I forgot to reply to the list too > >>> > >>> I used your code for more than one examples, and it works nicely. But > >> when > >>> I tried to use for the the function: f(x) = x^2, it looks like I am > >> missing > >>> something, but I could not figured it out. > >>> > >>> This what I used: > >>> > >>> > >>> > >>> f <- function(x) x^2 > >>> > >>> curve(f(x), from=-4, to=4, lwd=2, col="blue") > >>> abline(h=0) > >>> n <- 16 > >>> dx <- 8/n > >>> right <- (1:n)*dx > >>> left <- right - dx > >>> mid <- right - dx/2 > >>> fm <- f(mid) > >>> rect(left,0,right,fm, density = 20, border = "red") > >>> points(mid, fm, col = "red", cex = 1.25, pch=19) > >>> sum(fm*dx) > >>> > >>> > >>> > >>> 1/3 * (64+64) > >>> > >>> > >>> > >>> with many thanks > >>> steve > >>> > >>> On Fri, Nov 27, 2015 at 3:36 PM, Steven Stoline <sstoline at gmail.com> > >> wrote: > >>> > >>>> many thanks > >>>> > >>>> steve > >>>> > >>>> On Fri, Nov 27, 2015 at 9:20 AM, peter dalgaard <pdalgd at gmail.com> > >> wrote: > >>>> > >>>>> Something like this? > >>>>> > >>>>> f <- function(x) x^3-2*x > >>>>> curve(f(x), from=0, to=4) > >>>>> abline(h=0) > >>>>> n <- 16 > >>>>> dx <- 4/n > >>>>> right <- (1:n)*dx > >>>>> left <- right - dx > >>>>> mid <- right - dx/2 > >>>>> fm <- f(mid) > >>>>> points(mid, fm) > >>>>> rect(left,0,right,fm) > >>>>> > >>>>> sum(fm*dx) > >>>>> > >>>>> 1/4 * 4^4 - 4^2 > >>>>> > >>>>> > >>>>> -pd > >>>>> > >>>>> > >>>>> On 27 Nov 2015, at 13:52 , Steven Stoline <sstoline at gmail.com> > wrote: > >>>>> > >>>>>> Dear All: > >>>>>> > >>>>>> I am trying to explain to my students how to calculate the definite > >>>>>> integral using the Riemann sum. Can someone help me to graph the > area > >>>>> under > >>>>>> the curve of the function, showing the curve as well as the > >> rectangles > >>>>>> between 0 and 4.. > >>>>>> > >>>>>> *f(x) = x^3 - 2*x * > >>>>>> > >>>>>> over the interval [0 , 4] > >>>>>> > >>>>>> > >>>>>> > >>>>>> with many thanks > >>>>>> steve > >>>>>> > >>>>>> -- > >>>>>> Steven M. Stoline > >>>>>> 1123 Forest Avenue > >>>>>> Portland, ME 04112 > >>>>>> sstoline at gmail.com > >>>>>> > >>>>>> [[alternative HTML version deleted]] > >>>>>> > >>>>>> ______________________________________________ > >>>>>> R-help at r-project.org mailing list -- To UNSUBSCRIBE and more, see > >>>>>> https://stat.ethz.ch/mailman/listinfo/r-help > >>>>>> PLEASE do read the posting guide > >>>>> http://www.R-project.org/posting-guide.html > >>>>>> and provide commented, minimal, self-contained, reproducible code. > >>>>> > >>>>> -- > >>>>> Peter Dalgaard, Professor, > >>>>> Center for Statistics, Copenhagen Business School > >>>>> Solbjerg Plads 3, 2000 Frederiksberg, Denmark > >>>>> Phone: (+45)38153501 > >>>>> Office: A 4.23 > >>>>> Email: pd.mes at cbs.dk Priv: PDalgd at gmail.com > >>>>> > >>>>> > >>>>> > >>>>> > >>>>> > >>>>> > >>>>> > >>>>> > >>>>> > >>>>> > >>>> > >>>> > >>>> -- > >>>> Steven M. Stoline > >>>> 1123 Forest Avenue > >>>> Portland, ME 04112 > >>>> sstoline at gmail.com > >>>> > >>> > >>> > >>> > >>> -- > >>> Steven M. Stoline > >>> 1123 Forest Avenue > >>> Portland, ME 04112 > >>> sstoline at gmail.com > >>> > >>> [[alternative HTML version deleted]] > >>> > >>> ______________________________________________ > >>> R-help at r-project.org mailing list -- To UNSUBSCRIBE and more, see > >>> https://stat.ethz.ch/mailman/listinfo/r-help > >>> PLEASE do read the posting guide > >> http://www.R-project.org/posting-guide.html > >>> and provide commented, minimal, self-contained, reproducible code. > >> > > > > > > > > -- > > Steven M. Stoline > > 1123 Forest Avenue > > Portland, ME 04112 > > sstoline at gmail.com > > > > [[alternative HTML version deleted]] > > > > ______________________________________________ > > R-help at r-project.org mailing list -- To UNSUBSCRIBE and more, see > > https://stat.ethz.ch/mailman/listinfo/r-help > > PLEASE do read the posting guide > http://www.R-project.org/posting-guide.html > > and provide commented, minimal, self-contained, reproducible code. > > David Winsemius > Alameda, CA, USA > >-- Steven M. Stoline 1123 Forest Avenue Portland, ME 04112 sstoline at gmail.com [[alternative HTML version deleted]]
> On 16 Dec 2015, at 20:58 , Steven Stoline <sstoline at gmail.com> wrote: > > Dear David: > > > could you please try it for the functions f(x)=x^2 from -4 to 4 and the function f(x) = sqrt(x) from 0 to 4, and look watch the graphs.Looks fine to me, at least the non-plotmath version? -pd> > > Thank you very much for your helps. > > > steve > > > On Wed, Dec 16, 2015 at 2:09 PM, David Winsemius <dwinsemius at comcast.net> wrote: > > > On Dec 16, 2015, at 9:00 AM, Steven Stoline <sstoline at gmail.com> wrote: > > > > Dear William: *Left and Right Riemann Sums* > > > > > > Is there is a way to modify your function to compute Left Riemann Sum and > > Right Riemann Sum. I tried to modify yours, but i was not be able to make > > it work correctly. > > > > This is your function used to compute the Middle Riemann Sum. > > I think it's actually Dalgaard's method. > > > > showIntegral.med <- function (f, xmin, xmax, n = 16) > > { > > curve(f(x), from = xmin, to = xmax, lwd = 2, col = "blue") > > abline(h = 0) > > dx <- (xmax - xmin)/n > > right <- xmin + (1:n) * dx > > left <- right - dx > > mid <- right - dx/2 > > fm <- f(mid) > > rect(left, 0, right, fm, density = 20, border = "red") > > points(mid, fm, col = "red", cex = 1.25, pch = 19) > > sum(fm * dx) > > } > > > > > > > > ### Example 1: f(x) = x^2 , xmin=-4, xmax=4 > > ### ==============================> > > > > > > > showIntegral.med(f=function(x)x^2, xmin=-4, xmax=4, n=16) > > Wouldn't it just involve skipping the 'mid' calculations and using either the right or left values? Illustration for right: > > showIntegral.rt <- function (f, xmin, xmax, n = 16) > { > curve(f(x), from = xmin, to = xmax, lwd = 2, col = "blue") > abline(h = 0) > dx <- (xmax - xmin)/n > right <- xmin + (1:n) * dx > left <- right - dx > fr <- f(right) > rect(left, 0, right, fr, density = 20, border = "red") > points(right, fr, col = "red", cex = 1.25, pch = 19) > sum(fr * dx) > } > > You can make it prettier with plotmath: > > showIntegral.rt <- function (f, xmin, xmax, n = 16) > { > curve(f(x), from = xmin, to = xmax, lwd = 2, col = "blue") > abline(h = 0) > dx <- (xmax - xmin)/n > right <- xmin + (1:n) * dx > left <- right - dx > fr <- f(right) > rect(left, 0, right, fr, density = 20, border = "red") > points(right, fr, col = "red", cex = 1.25, pch = 19) > sum(fr * dx) > text(0,10, # might want to do some adaptive positioning instead > bquote( integral( .(body(f) )*dx, a, b) == .( sum(fr * dx )) ) ) > } > > -- > David. > > > > > > > > > with many thanks > > steve > > > > On Sat, Nov 28, 2015 at 1:11 PM, William Dunlap <wdunlap at tibco.com> wrote: > > > >> Your right <- (1:n)*dx mean that your leftmost rectangle's left edge > >> is at 0, but you want it to be at -4. You should turn this into a function > >> so you don't have to remember how the variables in your code depend > >> on one another. E.g., > >> > >> showIntegral <- function (f, xmin, xmax, n = 16) > >> { > >> curve(f(x), from = xmin, to = xmax, lwd = 2, col = "blue") > >> abline(h = 0) > >> dx <- (xmax - xmin)/n > >> right <- xmin + (1:n) * dx > >> left <- right - dx > >> mid <- right - dx/2 > >> fm <- f(mid) > >> rect(left, 0, right, fm, density = 20, border = "red") > >> points(mid, fm, col = "red", cex = 1.25, pch = 19) > >> sum(fm * dx) > >> } > >>> showIntegral(f=function(x)x^2, xmin=-4, xmax=4, n=16) > >> [1] 42.5 > >>> showIntegral(f=function(x)x^2, xmin=-4, xmax=4, n=256) > >> [1] 42.66602 > >>> showIntegral(f=function(x)x^2, xmin=-4, xmax=4, n=1024) > >> [1] 42.66663 > >> > >>> 2*4^3/3 > >> [1] 42.66667 > >>> showIntegral > >> Bill Dunlap > >> TIBCO Software > >> wdunlap tibco.com > >> > >> > >> On Fri, Nov 27, 2015 at 9:50 PM, Steven Stoline <sstoline at gmail.com> > >> wrote: > >>> Dear Peter: in my previous email I forgot to reply to the list too > >>> > >>> I used your code for more than one examples, and it works nicely. But > >> when > >>> I tried to use for the the function: f(x) = x^2, it looks like I am > >> missing > >>> something, but I could not figured it out. > >>> > >>> This what I used: > >>> > >>> > >>> > >>> f <- function(x) x^2 > >>> > >>> curve(f(x), from=-4, to=4, lwd=2, col="blue") > >>> abline(h=0) > >>> n <- 16 > >>> dx <- 8/n > >>> right <- (1:n)*dx > >>> left <- right - dx > >>> mid <- right - dx/2 > >>> fm <- f(mid) > >>> rect(left,0,right,fm, density = 20, border = "red") > >>> points(mid, fm, col = "red", cex = 1.25, pch=19) > >>> sum(fm*dx) > >>> > >>> > >>> > >>> 1/3 * (64+64) > >>> > >>> > >>> > >>> with many thanks > >>> steve > >>> > >>> On Fri, Nov 27, 2015 at 3:36 PM, Steven Stoline <sstoline at gmail.com> > >> wrote: > >>> > >>>> many thanks > >>>> > >>>> steve > >>>> > >>>> On Fri, Nov 27, 2015 at 9:20 AM, peter dalgaard <pdalgd at gmail.com> > >> wrote: > >>>> > >>>>> Something like this? > >>>>> > >>>>> f <- function(x) x^3-2*x > >>>>> curve(f(x), from=0, to=4) > >>>>> abline(h=0) > >>>>> n <- 16 > >>>>> dx <- 4/n > >>>>> right <- (1:n)*dx > >>>>> left <- right - dx > >>>>> mid <- right - dx/2 > >>>>> fm <- f(mid) > >>>>> points(mid, fm) > >>>>> rect(left,0,right,fm) > >>>>> > >>>>> sum(fm*dx) > >>>>> > >>>>> 1/4 * 4^4 - 4^2 > >>>>> > >>>>> > >>>>> -pd > >>>>> > >>>>> > >>>>> On 27 Nov 2015, at 13:52 , Steven Stoline <sstoline at gmail.com> wrote: > >>>>> > >>>>>> Dear All: > >>>>>> > >>>>>> I am trying to explain to my students how to calculate the definite > >>>>>> integral using the Riemann sum. Can someone help me to graph the area > >>>>> under > >>>>>> the curve of the function, showing the curve as well as the > >> rectangles > >>>>>> between 0 and 4.. > >>>>>> > >>>>>> *f(x) = x^3 - 2*x * > >>>>>> > >>>>>> over the interval [0 , 4] > >>>>>> > >>>>>> > >>>>>> > >>>>>> with many thanks > >>>>>> steve > >>>>>> > >>>>>> -- > >>>>>> Steven M. Stoline > >>>>>> 1123 Forest Avenue > >>>>>> Portland, ME 04112 > >>>>>> sstoline at gmail.com > >>>>>> > >>>>>> [[alternative HTML version deleted]] > >>>>>> > >>>>>> ______________________________________________ > >>>>>> R-help at r-project.org mailing list -- To UNSUBSCRIBE and more, see > >>>>>> https://stat.ethz.ch/mailman/listinfo/r-help > >>>>>> PLEASE do read the posting guide > >>>>> http://www.R-project.org/posting-guide.html > >>>>>> and provide commented, minimal, self-contained, reproducible code. > >>>>> > >>>>> -- > >>>>> Peter Dalgaard, Professor, > >>>>> Center for Statistics, Copenhagen Business School > >>>>> Solbjerg Plads 3, 2000 Frederiksberg, Denmark > >>>>> Phone: (+45)38153501 > >>>>> Office: A 4.23 > >>>>> Email: pd.mes at cbs.dk Priv: PDalgd at gmail.com > >>>>> > >>>>> > >>>>> > >>>>> > >>>>> > >>>>> > >>>>> > >>>>> > >>>>> > >>>>> > >>>> > >>>> > >>>> -- > >>>> Steven M. Stoline > >>>> 1123 Forest Avenue > >>>> Portland, ME 04112 > >>>> sstoline at gmail.com > >>>> > >>> > >>> > >>> > >>> -- > >>> Steven M. Stoline > >>> 1123 Forest Avenue > >>> Portland, ME 04112 > >>> sstoline at gmail.com > >>> > >>> [[alternative HTML version deleted]] > >>> > >>> ______________________________________________ > >>> R-help at r-project.org mailing list -- To UNSUBSCRIBE and more, see > >>> https://stat.ethz.ch/mailman/listinfo/r-help > >>> PLEASE do read the posting guide > >> http://www.R-project.org/posting-guide.html > >>> and provide commented, minimal, self-contained, reproducible code. > >> > > > > > > > > -- > > Steven M. Stoline > > 1123 Forest Avenue > > Portland, ME 04112 > > sstoline at gmail.com > > > > [[alternative HTML version deleted]] > > > > ______________________________________________ > > R-help at r-project.org mailing list -- To UNSUBSCRIBE and more, see > > https://stat.ethz.ch/mailman/listinfo/r-help > > PLEASE do read the posting guide http://www.R-project.org/posting-guide.html > > and provide commented, minimal, self-contained, reproducible code. > > David Winsemius > Alameda, CA, USA > > > > > -- > Steven M. Stoline > 1123 Forest Avenue > Portland, ME 04112 > sstoline at gmail.com-- Peter Dalgaard, Professor, Center for Statistics, Copenhagen Business School Solbjerg Plads 3, 2000 Frederiksberg, Denmark Phone: (+45)38153501 Office: A 4.23 Email: pd.mes at cbs.dk Priv: PDalgd at gmail.com