Trouble converting definite integrals to Riemann's and back

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Thread starter Terrell
Start date Apr 21, 2016
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Definite integrals Integrals

In summary, Riemann's integral is a method for calculating the area under a curve on a given interval and is important in mathematics for its ability to calculate areas and volumes for irregularly shaped objects. To convert a definite integral to Riemann's integral, the interval must be divided into smaller subintervals and the area under the curve must be approximated using rectangles. Some common challenges in this conversion include determining the appropriate number of subintervals and choosing the correct width and height of the rectangles. Riemann's integral can only be used for continuous functions and can be converted back to a definite integral by taking the limit as the number of subintervals approaches infinity. Other methods, such as Lebesgue integration, may be

Apr 21, 2016

Terrell

can i request anyone to please show me the step by step with specific explanations? thank you! i saw this on stackexchange, and the steps shown are really fuzzy to me :(

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Apr 21, 2016

mathman

Science Advisor

In general [itex]\int_0^1 f(x)dx [/itex] can be approximated by a Riemann sum [itex]\frac{1}{n}\sum_{k=1}^n f(\frac{k-1/2}{n})[/itex]. For large n, -1/2 can be ignored.

1. What is Riemann's integral and why is it important in mathematics?

Riemann's integral is a method for calculating the area under a curve on a given interval. It is important in mathematics because it allows for the calculation of areas and volumes for irregularly shaped objects, which is crucial in many areas of science and engineering.

2. How do you convert a definite integral to Riemann's integral?

To convert a definite integral to Riemann's integral, you must first divide the interval into smaller subintervals and then approximate the area under the curve on each subinterval using rectangles. As the number of subintervals increases, the approximation becomes more accurate and approaches the true value of the integral.

3. What are some common challenges in converting definite integrals to Riemann's integral?

One common challenge is determining the appropriate number of subintervals to use for the approximation. Too few subintervals can result in a large error, while too many subintervals can make the calculation computationally difficult. Another challenge is choosing the correct width and height of the rectangles to ensure an accurate approximation.

4. Can Riemann's integral be used for any type of function?

Riemann's integral can be used for continuous functions, but it may not be applicable to discontinuous or non-smooth functions. In these cases, other methods such as Lebesgue integration may be more appropriate.

5. How can I convert Riemann's integral back to a definite integral?

To convert Riemann's integral back to a definite integral, you must take the limit as the number of subintervals approaches infinity. This will give the exact value of the integral, which can then be used in the original definite integral expression.