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Definite Integrals


Date: 05/04/97 at 19:22:30
From: Joseph Padlo
Subject: Integrals

What is the difference between "the area under the curve from a to b"
and "the definite integral from a to b"?

jp


Date: 05/05/97 at 08:44:45
From: Doctor Jerry
Subject: Re: Integrals

Hi Joseph,

This is a really good question!

The definite integral of a continuous function on an interval [a,b] is 
defined in terms of a limit of approximating sums. An example of an 
approximating sum is to subdivide the interval [a,b] into n equal 
subintervals with the points (I'll use x_0, x_1, etc, for x sub 0, x 
sub 1, etc) a = x_0 < x_1< x2_ <...< x_n = b and then calculate the 
minimum m_i and maximum M_i of f on the subinteval from x_{i-1} to 
x_i. Letting h = (b-a)/n, the sums L_n = h(m_1+m_2+...+m_n) and
U_n = h(M_1+M_2+...+M_n) are lower and upper sums that approximate the 
definite integral of f on [a,b].  As n becomes large, these lower and 
upper sums will approach a common limit.  This is the definite 
integral.

One interpretation of the definite integral of f on [a,b] is the area 
under the graph of f and above the inteval [a,b].  Assuming f is non-
negative on [a,b], the lower and upper sums correspond to inscribed 
and circumscribed rectangles, thus making a intuitively clear 
connection between the definite integral and area.

Other interpretations of the definite integral of f on [a,b] are the 
mass of a rod with variable density, probability calculations, arc 
length, volumes of revolution, etc.

In sum, the area under the curve from a to b is just one of many 
interpretations of the definite integral.


-Doctor Jerry,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   


Date: 05/19/97 at 20:03:42
From: Joseph Padlo
Subject: Integrals

I have asked this question before, but I don't think I phrased it very 
well, so I would like to try again.

What is the difference between the area under the curve f(x) from A to
B and the integral of f(x) from A to B?

Let's take a specific example: let f(x) = x; A = -2, B = 2. It is 
obvious that the area between -2 and 0 is negative. Is the integral 
from negative 2 to positive 2 of x equal to 0 or 4?

This problem has been bothering me a long time. I read a problem in a
book that hinted that there was a difference, but I loaned the book to
someone and it was not returned.

J.P.


Date: 05/21/97 at 16:38:11
From: Doctor Anthony
Subject: Re: integrals

Joseph,

The answer is zero if you want the strict algebraic result.

The area is calculated from: INT(a to b)[y.dx]

If y is negative (i.e. below the x axis), then the area will be 
negative.  If you integrate between two limits which have equal areas 
below and above the x axis (as in the example you quoted), then the 
TOTAL area will be zero.  If you want to find the magnitude of the 
area on each side of the x axis, then you must integrate between 
intermediate values of x (i.e. to the point where the curve crosses 
the x axis).

-Doctor Anthony,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Calculus

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