Definite IntegralsDate: 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/ |
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