Integrate sqrt(1+(1\sqrt(x)))^2

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July 15th 2010, 06:05 AM
MechEng

Integrate sqrt(1+(1\sqrt(x)))^2

Good morning,

I left off on this problem yesterday. I am given:

Find the surface area by revolving around the x-axis:
$f(x) = 2\sqrt{x}$ for $1\leq x \leq4$

So far I have the following:

$<br /> f'(x) = \frac{1}{\sqrt{x}}<br />$

$\sqrt{1+(\frac{1}{\sqrt{x}})^2} = \sqrt{\frac{x+1}{x}}$

So...

$<br /> L = \int_{1}^{4}{\sqrt{\frac{x+1}{x}} dx<br />$

And... I'm stuck.

I don't now why roots are giving me so much trouble lately. Is there something that I have forgotten or seem to be missing? I don't recall roots being this problematic.
July 15th 2010, 06:22 AM
Ted

The problems wants you to find the surface area not the length of the curve.
July 15th 2010, 07:01 AM
MechEng

Yes, it does want me to find the surface area. However, I still need to be able to integrate $\int_{1}^{4}{\sqrt{\frac{x+1}{x}} dx$ as $S=\int_{1}^{4}{2\pi\sqrt{\frac{x+1}{x}} dx$ is the same as $S=2\pi\int_{1}^{4}{\sqrt{\frac{x+1}{x}} dx$ ... No?

Is finding the surface not as simple as multiplying the lentgh of the curve by $2\pi$ ?
July 15th 2010, 08:25 AM
Ted

Quote:

Originally Posted by MechEng

Is finding the surface not as simple as multiplying the lentgh of the curve by $2\pi$ ?

Becareful, you are multiplying the dS (which is $\sqrt{1+f^2(x)}$ ) not the length of the curve!

For your question, you will multiply the dS by $2\pi y$ if the revolution about the x-axis.
and you will multiply it by $2\pi x$ if the revolution about the y-axis.

So the required surface area is:

$4\pi \int_1^4 \sqrt{x} \, \sqrt{\frac{x+1}{x}} \, dx$

Can you solve it?
July 15th 2010, 09:14 AM
MechEng

I think so...

Jumping to the end, I get:

$S = \frac{8\pi(5\sqrt{5}-2\sqrt{2})}{3}$

Does that look correct to you?
July 15th 2010, 09:38 AM
Ted

That 8 should be 4.
Let see your work.
July 15th 2010, 10:01 AM
MechEng

$S=\int_{1}^{4}{2\pi2\sqrt{x}\sqrt{\frac{x+1}{x}}} dx$

$S=4\pi\int_{1}^{4}{\sqrt{x}\sqrt{\frac{x+1}{x}}} dx$

$S=4\pi\int_{1}^{4}{x^{\frac{1}{2}}(x+1)^{\frac{1}{ 2}}x^{-\frac{1}{2}}} dx$

$S=4\pi\int_{1}^{4}{(x+1)^{\frac{1}{2}}} dx$
July 15th 2010, 10:09 AM
Ted

Good. Now, substitute $u=x+1$ .
Lets see what did you get.
July 15th 2010, 10:09 AM
MechEng

$S=4\pi\frac{2(x+1)^{3/2}}{3}$ $|_{1}^{4}$

$S=\frac{8\pi(x+1)^{3/2}}{3}$ $|_{1}^{4}$

$S=\frac{8\pi(4+1)^{3/2}}{3}-\frac{8\pi(1+1)^{3/2}}{3}$

$S=\frac{8\pi(5)^{3/2}}{3}-\frac{8\pi(2)^{3/2}}{3}$

$S=\frac{8\pi5\sqrt{5}}{3}-\frac{8\pi2\sqrt{2}}{3}$

$S=\frac{8\pi(5\sqrt{5}-2\sqrt{2})}{3}$

... Did I miss something?
July 15th 2010, 10:20 AM
Ted

No. You are correct.
I forgot that added "2".
I considered it as 2 pi not 4 pi.
Sorry for that.
Good Luck.