| 1. | It is popular to define the Riemann integral as the Darboux integral.
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| 2. | The Riemann integral can only integrate functions on a bounded interval.
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| 3. | If you use the Riemann integral, the answer is simply yes.
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| 4. | A better route is to abandon the Riemann integral for the Lebesgue integral.
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| 5. | As the shapes get smaller and smaller, the sum approaches the Riemann integral.
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| 6. | The most commonly used definitions of integral are Riemann integrals and Lebesgue integrals.
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| 7. | But this is a fact that is beyond the reach of the Riemann integral.
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| 8. | There are some other technical difficulties with the Riemann integral.
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| 9. | The Riemann integral uses the notion of length explicitly.
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| 10. | The definition of a Banach-valued Riemann integral is an evident modification of the usual one.
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