the definite integral as a sum

4
Take a look at Examples 1 and 3 in the text. Example 1 Total Cost Your cellular phone company ofers you an innovative pricing scheme. When you make a call, the marginal co o the t th minute o the call !ill "e c(t) = 20/(t + 100) $/min. #se a numerical calculation to estimate the cost o a $% minute phone call. Solution &s !e see in the text, the total cost is given "y the de'nite integral, (omputing this integral numerically amounts to "reaking the interval )%, $%* into a large num"er o short intervals, and adding up the costs over these intervals. The smaller the interval, the more accurate the Thus, let us "egin "y deciding on the num"er o divisions !e !ill use, and setting up all the values o t n 1+% The num"er o su"divisions a % The start o the time interval lo!er limit o integration- " $% The end o the time interval upper limit o integration- elta t %./ t c t- c t-0 elta t iemann 2um t starts at a. % &dd elta t 1. We !ill 'll in the t column !it at each step. and continuing in steps o el We !ould like to "e a"le to c length o the columns each ti 5nstead o !orrying a"out sto let t go on or 1%%% or so step you are interested in-, and th !e don7t !ant to sum terms c 2. Test your ormulas "y changi 8o do!n or 9o! 'll in the ormula or c t- 1%%% or so =20/(t+100 steps. !e !ill use =IF(t<$$2 This sets c t- +%; t<1%%- i to ?ero other!ise. 2ee the tu The !idth o each su"interval, given "y " a-;n look at its ormula do not ch 0 60 20 t +100 dt

Upload: mightsright

Post on 04-Nov-2015

15 views

Category:

Documents


0 download

DESCRIPTION

The Definite Integral as a Sum in Excel

TRANSCRIPT

Example 1Take a look at Examples 1 and 3 in the text.Example 1Total CostYour cellular phone company offers you an innovative pricing scheme. When you make a call, the marginal costof the t-th minute of the call will bec(t) = 20/(t + 100) $/min.Use a numerical calculation to estimate the cost of a 60 minute phone call.SolutionAs we see in the text, the total cost is given by the definite integral,Computing this integral numerically amounts to breaking the interval [0, 60] into a large number of shortintervals, and adding up the costs over these intervals. The smaller the interval, the more accurate the answer.Thus, let us begin by deciding on the number of divisions we will use, and setting up all the values of t.n120The number of subdivisionsa0The start of the time-interval (lower limit of integration)b60The end of the time-interval upper limit of integration)Delta t0.5The width of each subinterval, given by (b-a)/n (look at its formula -- do not change this value)tc(t)c(t)*Delta tRiemann Sumt starts at a.0Add Delta t1.We will fill in the t-column with values starting at a = 0.at each step.and continuing in steps of Delta t.We would like to be able to change n without changing thelength of the columns each time, so we will use a trick:Instead of worrying about stopping when you reach b,let t go on for 1000 or so steps (or the largest value of nyou are interested in), and then set c(t) = 0 if t >= b, sincewe don't want to sum terms c(t) if t >= b (see the text).2.(Test your formulas by changing n, a, and b.)Go down forNow fill in the formula for c(t). However, instead of using1000 or so=20/(t+100)steps.we will use=IF(t