Indefinite and Definite Integrals of ZeroDate: 09/12/2004 at 13:56:15 From: Vando Subject: Integration Hello Dr. Math, my question is about the integral of zero. It's a constant, because the derivative of a constant is zero, right? But how can this be true, because I also was taught that an integral is the area under a curve...and there is no area under the line x = 0, right? Can you help me understand this seeming paradox? Date: 09/13/2004 at 03:21:01 From: Doctor Luis Subject: Re: Integration Hi Vando, There are two types of integrals at play here. Definite integrals are the ones that describe the actual area under a curve. Indefinite integrals are the ones that describe the anti-derivative. There's no paradox, really. When speaking of indefinite integrals, the integral of 0 is just 0 plus the usual arbitrary constant, i.e., / | | 0 dx = 0 + C = C | / There's no contradiction here. When evaluating the area under a curve f(x), we find the antiderivative F(x) and then evaluate from a to b: / b | | f(x) dx = F(b) - F(a) | / a So, for f(x) = 0, we find F(x) = C, and so F(b) - F(a) = C - C = 0. Thus, the total area is zero, as we expected. Does it make sense now? Let us know if you have any more questions. - Doctor Luis, The Math Forum http://mathforum.org/dr.math/ Date: 09/13/2004 at 15:56:09 From: Vando Subject: Thank you (Integration) Thank you for making the two types of integral definitions clear! :) |
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