One way to compute an integral
In this unit we will learn to compute an estimate of an integral in a new way, and we will compare the result to the theoretical computation of the same integral, and to the integral as computed by fnInt.
The general case: Evaluating an integral from A to B.
Randomly choose a number between A and B, with A<B.
(Remember that RAND on the TI-83/84 yields a random number between 0 and 1.)
To get a random number between A and B, enter
A + RAND*(B-A)
(RAND yields a number between 0 and 1. If it yields 0, the above yields A. If it yields 1, the above yields B.)
RAND(100) yields 100 random numbers between 0 and 1.
A + RAND(100)*(B-A) yields 100 random numbers between A and B.
Put a function in Y1.
Now enter from the home screen
sum(Y1(A+RAND(100)*(B-A)))/100
You are computing the average of 100 values of Y1(X), and multiplying it by (B-A).
This will give an estimate of the value of the integral! The integral is the average of the values of the function times (B-A).
So in general, the integral divided by the range of x is the average. When the range of x is one, then the integral is the average.
The specific case: Evaluating an integral from 0 to 1 (so the range of x is one).
To simplify, let’s just consider values between 0 and 1. (So here, A = 0 and B =1, so B-A=1.)
Let our first function be y = x2. Put it in Y=:
\Y1=X2
Now enter from the home screen
sum(Y1(RAND(100)))/100
You are randomly dividing the interval from 0 to 1 into 100 pieces, which are little dx’s.
You evaluate the function at each of these values of x, and you sum them up and divide by 100.
Display: .343816717 (you won’t get exactly this, because your random numbers will be different!)
We can compute
And we can compute
fnInt(X2,X,0,1)
display:.3333333333
You may increase the number of random numbers you use, up to 999! But then you must be very patient.
Let’s try another integral:
y = 1/(1+x).
You may recognize this as the natural logarithm of 2.
ln(2) = .6931471806
fnInt((1+X)-1,X, 0, 1)
display .6931471806
Now put
Y1=(1+X)-1
From the home screen, enter
sum(Y1(RAND(100)))/100
I got this: .7066311345
You may generate up to 999 values for RAND, but again you must be very patient.
Calculus Index