Limit of the Integral Sum

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To integrate functions as the limit of an integral sum, how can we know which way to take the partition points in the interval?

For example, in
$$\int_{a}^{b} x dx$$

I can take the partition points as

$$x_o = a$$
$$x_1 = a + \delta x$$
...
$$x_k = a+ k\delta x$$
where $$\delta x = \frac{b-a}{n}$$

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

But to integrate
$$\int_{a}^{b} \sqrt{x} dx$$

If I take the partition points as above the sum will be
$$(\delta x)(\sqrt{a} + \sqrt{a+ \delta x} + ... )$$

which I cannot find.

I can solve the question if I take the partition points as
$$x_0 = a$$
$$x_1 = aq$$
...
$$x_k=aq^k$$

Where $$q=(\frac{b}{a})^(1/n)$$.
{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?

matt grime
Homework Helper
the choice of partition depends upon the function. experience is the thing you need and an educated guess. Ideally you want to choose the partition such that the resulting series is one you can sum. egn if tou wre doing sqrt(x) from 0 to 1 then picking the partition i^2/n^2 as i goes from 0 to n is good cos the square root spits out i/n.

Last edited:
HallsofIvy