Understanding the Differences between Integrating and Taking the Square Root

[iScience]

$$\int \sqrt{f(x)}dx$$ is NOT the same as..

$$\sqrt{\int f(x)dx}$$ right?

i did an example problem and they turned out not to be equivalent but i just wanted to make certain.




if they actually aren't equivalent.. why aren't they?

 

[iScience]

well, better example:

$$\sqrt{(C\int\sqrt{(a+b)}dx)^2}$$

this is not the same as..

$$\sqrt{C^2\int(a+b)dx}$$

is it?

(C= constant, a= variable, b=constant (not that it really matters..))

 

[PeroK]

$$\int \sqrt{f(x)}dx$$ is NOT the same as..

$$\sqrt{\int f(x)dx}$$ right?

i did an example problem and they turned out not to be equivalent but i just wanted to make certain.




if they actually aren't equivalent.. why aren't they?

Try ##f(x)=x^2## and you'll see for yourself why they are not the same.

 

[gopher_p]

Remember that the definite integral is a kind of sum (technically a limit of sums), and so one way to understand why things may or may not work with the integral is to compare it to a sum.

In this case, if you remember that ##\sqrt{a+b}\neq\sqrt{a}+\sqrt{b}## most of the time, and in general ##\sqrt{\sum_1^n a_i}\neq\sum_1^n\sqrt{a_i}##, it's not too hard to see why, most of the time, ##\sqrt{\int_a^bf(x)\ dx}\neq\int_a^b\sqrt{f(x)}\ dx##.

 

[iScience]

Remember that the definite integral is a kind of sum (technically a limit of sums), and so one way to understand why things may or may not work with the integral is to compare it to a sum.

In this case, if you remember that ##\sqrt{a+b}\neq\sqrt{a}+\sqrt{b}## most of the time, and in general ##\sqrt{\sum_1^n a_i}\neq\sum_1^n\sqrt{a_i}##, it's not too hard to see why, most of the time, ##\sqrt{\int_a^bf(x)\ dx}\neq\int_a^b\sqrt{f(x)}\ dx##.

Thanks this was helpful! :)