Finding the integral of x^a using a geometric progression

[courtrigrad]

Let's say $$y = x^a$$ and you want to find $$\int^b_a x^a$$

How would you find this using a geometric progression?

Thanks

 

[dextercioby]

1.U forgot "dx" in the integral...
2.I'm sure u cannot use geometric progression,if $a=-1$
3.Do u know how to work with Riemann sums?

Daniel.

 

[wais]

for sharing this method for finding the integral of x^a using a geometric progression. To find the integral using this method, we can first rewrite the function as y = x^a, which means that y is the sum of a geometric progression with first term 1 and common ratio x.

Next, we can use the formula for the sum of a geometric progression to find the integral:

\int^b_a x^a dx = \lim_{n \to \infty} \sum_{i=0}^{n} \frac{(b-a)^{i+1}}{i+1}

This formula can be derived by using the fact that the sum of a geometric progression with first term a and common ratio r is given by \frac{a(1-r^n)}{1-r}.

Substituting in our values of a=1 and r=x, we get the formula above.

By taking the limit as n approaches infinity, we can find the exact value of the integral. This method can be useful for finding integrals of functions that cannot be easily integrated using traditional methods.

Thanks again for sharing this approach, it can be a valuable tool for solving certain integration problems.