Epsilon Posted July 19, 2002 Share Posted July 19, 2002 Okay, I am in the middle of learning single variable calculus and I can understand most of the topics I've learned so far. There are some things that are not obvious or easy understand to me, e.g. the Fundamental Theorem of Calculus: int(a,b, f(x) dx) = F(b) - F(a) , where F is an antiderivative of f. I can understand that in order to find the area between two intervals one would subtract the smaller area from the larger. So I guess this would imply that taking the antiderivative of a function and evaluating it at some point (we'll call it b) would give you the area from 0 to b. (Someone verify this) I see computing the area under a curve from the development of the Riemann Sum, but how is that the same as evaluating the difference between two antiderivatives ( F(b), and F(a) ) ? If anyone can give a nice explanation or reference on FTC I would appreciate it. Sorry for any typos. Thanks. Link to comment Share on other sites More sharing options...
fafalone Posted July 19, 2002 Share Posted July 19, 2002 The fundamental theorem is just saying the area under a curve, from a to b, is equal to the integrals value with b minus a Take f(x) = x F(x) = (1/2)x^2 If we want to find the area under y=x from x=0 to x=1, we take (1/2)1^2 = 1/2 and subtract (1/2)0^2=0 So, the area is 1/2 Look at f(x) as the slope at that point, so you find the 'original' equation. A Riemann sum is an approximation. Some functions cannot be integrated, such as x^x. These must be approximated using end-points and the mid-point to estimate the area. This is usually not an exact value, unlike normal integration. Link to comment Share on other sites More sharing options...
Epsilon Posted July 19, 2002 Author Share Posted July 19, 2002 Right, I know that one can compute the area under a curve using FTC. However I don't see the connection by just subtracting the two antiderivatives. It seems like you should just make a different notation for antidifferentation and a different notation for computing definite integrals ... Sorry about that. I should have said the limit of a riemann sum as the number of intervals approach infinity. (Which would yield an exact answer assuming you can find its limit [whether or not the sums are left, right, or mid-point sums]) Link to comment Share on other sites More sharing options...
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