Bernhard Riemann: Difference between revisions
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In the Riemann integral, you partition the domain into equal pieces, and compute the area of the corresponding rectangles. For the upper Riemann sum, the height of each rectaingle is the largest value of the function on each piece of the domain. For the lower Riemann sum, use the smallest value of the function on each piece of the domain. As you decrease the size of the pieces of the domain to zero, if the upper sum and lower sum converge to the same number, the function is Riemann integrable. Every continuous function is Riemann integrable, but some wildly discontinuous functions are not. For example, let f(x) = 0 when x is irrational, 1 when x is rational. The Lebesgue integral solves this problem by partitioning the range rather than the domain. Every Riemann integrable function gives the same answer when you compute its Lebesgue integral. |
In the Riemann integral, you partition the domain into equal pieces, and compute the area of the corresponding rectangles. For the upper Riemann sum, the height of each rectaingle is the largest value of the function on each piece of the domain. For the lower Riemann sum, use the smallest value of the function on each piece of the domain. As you decrease the size of the pieces of the domain to zero, if the upper sum and lower sum converge to the same number, the function is Riemann integrable. Every continuous function is Riemann integrable, but some wildly discontinuous functions are not. For example, let f(x) = 0 when x is irrational, 1 when x is rational. The Lebesgue integral solves this problem by partitioning the range rather than the domain. Every Riemann integrable function gives the same answer when you compute its Lebesgue integral. |
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''Shouldn't this article live at [[Riemann integral]]?'' |
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Revision as of 18:46, 12 December 2001
In the Riemann integral, you partition the domain into equal pieces, and compute the area of the corresponding rectangles. For the upper Riemann sum, the height of each rectaingle is the largest value of the function on each piece of the domain. For the lower Riemann sum, use the smallest value of the function on each piece of the domain. As you decrease the size of the pieces of the domain to zero, if the upper sum and lower sum converge to the same number, the function is Riemann integrable. Every continuous function is Riemann integrable, but some wildly discontinuous functions are not. For example, let f(x) = 0 when x is irrational, 1 when x is rational. The Lebesgue integral solves this problem by partitioning the range rather than the domain. Every Riemann integrable function gives the same answer when you compute its Lebesgue integral.
Shouldn't this article live at Riemann integral?