To integrate functions as the limit of an integral sum, how can we know which way to take the partition points in the interval?(adsbygoogle = window.adsbygoogle || []).push({});

For example, in

[tex] \int_{a}^{b} x dx [/tex]

I can take the partition points as

[tex] x_o = a [/tex]

[tex] x_1 = a + \delta x [/tex]

...

[tex] x_k = a+ k\delta x[/tex]

where [tex] \delta x = \frac{b-a}{n} [/tex]

So that the sum is [tex] \sum_{k=1}^{n} f(x_{k-1}) \delta x [/tex]

But to integrate

[tex] \int_{a}^{b} \sqrt{x} dx [/tex]

If I take the partition points as above the sum will be

[tex] (\delta x)(\sqrt{a} + \sqrt{a+ \delta x} + ... ) [/tex]

which I cannot find.

I can solve the question if I take the partition points as

[tex] x_0 = a [/tex]

[tex] x_1 = aq [/tex]

...

[tex] x_k=aq^k [/tex]

Where [tex] q=(\frac{b}{a})^(1/n) [/tex].

{The idea to take it this way was given as a hint in the book}

So, is there any other specific manner in which I should spilt the partition points and if so is there a general method in which I can know how to take the values of x_0, x_1, x_2 to solve the problem?

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# Limit of the Integral Sum

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