Need help simplifying 2pi int[0,b]|a sqrt{x}|*sqrt{1+((a^2)/(4x))} dx

[dyzh]

Can someone please help me simplify this?

. . . . .\(\displaystyle \displaystyle \large{2\pi\, \int_0^b\, \Bigg|\, a\, \sqrt{\strut x\,}\, \Bigg|\, \cdot\, \sqrt{\strut 1\, +\, \dfrac{a^2}{4x}\,}\, dx}\)

Thank you!

 

[Deleted member 4993]

Can someone please help me simplify this?

. . . . .\(\displaystyle \displaystyle \large{2\pi\, \int_0^b\, \Bigg|\, a\, \sqrt{\strut x\,}\, \Bigg|\, \cdot\, \sqrt{\strut 1\, +\, \dfrac{a^2}{4x}\,}\, dx}\)

Thank you!

. . . . .\(\displaystyle \displaystyle \large{2\pi\, \int_0^b\, \Bigg|\, a\, \sqrt{\strut x\,}\, \Bigg|\, \cdot\, \sqrt{\strut 1\, +\, \dfrac{a^2}{4x}\,}\, dx}\)

What are your thoughts regarding the assignment?

Please share your work with us ...even if you know it is wrong.

If you are stuck at the beginning tell us and we'll start with the definitions.

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[dyzh]

It's a part of my math project. I need a simplified version of this equation so I can substitute numbers and apply it for different situations. But I'm stuck here so I can't proceed forward.

 

[tkhunny]

Can someone please help me simplify this?

. . . . .\(\displaystyle \displaystyle \large{2\pi\, \int_0^b\, \Bigg|\, a\, \sqrt{\strut x\,}\, \Bigg|\, \cdot\, \sqrt{\strut 1\, +\, \dfrac{a^2}{4x}\,}\, dx}\)

Thank you!

1) It's not an "equation".
2) is "a" ever negative?

 

[Deleted member 4993]

Try substituting:

x = (1/4)*(a^2)*[cot(Θ)]^2

 

[dyzh]

a and b are both positive

 

[dyzh]

Try substituting:

x = (1/4)*(a^2)*[cot(Θ)]^2

how? and why?

 

[tkhunny]

a and b are both positive

Well, then what's the point of the Absolute Values?

 

[Deleted member 4993]

how? and why?

How - Are you saying that you do not know method of substitution in a function?

Why - because that will make your integrand " simpler (you wanted to simplify - didn't you?)

 

[Dr.Peterson]

Can someone please help me simplify this?

. . . . .\(\displaystyle \displaystyle \large{2\pi\, \int_0^b\, \Bigg|\, a\, \sqrt{\strut x\,}\, \Bigg|\, \cdot\, \sqrt{\strut 1\, +\, \dfrac{a^2}{4x}\,}\, dx}\)

Thank you!

Let's back up a bit. What you are asking for is to integrate a function -- specifically, to evaluate a definite integral. It would be very helpful if you told us how much you know about calculus, so we would have a better idea where you need help. It would also help if you gave some context.

Since you have said that a and b are both known to be positive, we don't need the absolute value (because |a| = a for positive numbers), and we can just write this:

\(\displaystyle \displaystyle \large{2\pi\, \int_0^b\, a\, \sqrt{x\,}\, \sqrt{1\, +\, \dfrac{a^2}{4x}\,}\, dx}\)

The substitution that was suggested was based on the form \(\displaystyle \displaystyle \sqrt{1\, +\, \dfrac{a^2}{4x}\,} = \sqrt{1\, +\, u^2\,}\), which is reminiscent of the fact that \(\displaystyle 1 + \tan^2 \theta = \sec^2 \theta\). The fact that x is not squared makes this somewhat different from the usual trig substitution, but if you try it, everything will (eventually) work out nicely. (A lot of integration is a matter of seeing a possibility and just trying it in case it will work.)

I myself saw a slightly different approach, taking smaller steps. We can first combine the two radicals into

\(\displaystyle \displaystyle \large{2\pi\, \int_0^b\, a\, \sqrt{x\left(1\, +\, \dfrac{a^2}{4x}\,\right)}\, dx = 2\pi\, a\, \int_0^b\, \sqrt{x\, +\, \dfrac{a^2}{4}}\, dx}\)

Then I did a similar substitution, and if I did the work right, this worked too.

Are you able to continue from here? If not, where do you have trouble? Have you ever used trig substitution? (If you just need an answer, WolframAlpha can do that for you, though it doesn't always simplify as much as you'd want.)