Stumped on an integral.

**Malaclypse** · March 8th 2011, 10:11 PM

Been beating my head against the wall on an integral for awhile now....

The question is this:

Compute the surface area generated by revolving the curve $f(x) = 2\sqrt x ,\,\,{\text{for 1}} \leqslant x \leqslant 4$ about the x-axis.

I took the 1st derivative of the equation:

$f'(x) = \frac{1}{x}$

Easy enough. Then I went into my integral:

$2\pi \int_{\,1}^{\,4} {2\sqrt x \sqrt {1 + \frac{1}{{{x^2}}}} } dx$

That didn't look very pretty, and I didn't seem to be getting anywhere trying to solve it (no u-sub, and integration by parts didn't seem to be working), so I tried to manipulate the equation several times to see if a different variation would work...

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

and

$4\pi \int_{\,1}^{\,4} {\left( {\sqrt {\frac{1}{x}} } \right)} \sqrt {\left( {{x^2} + 1} \right)} dx$

and

$4\pi \int_{\,1}^{\,4} {\left( {\sqrt {\frac{1}{x}} } \right)} \sqrt {{x^2}\left( {1 + \frac{1}{{{x^2}}}} \right)} dx$

and

$4\pi \int_{\,1}^{\,4} {x\sqrt {\left( {\frac{{{x^2} + 1}}{{{x^3}}}} \right)} } dx$

But I can't find one to work out. My professor stated on the assignment that the integral can be done exactly, otherwise I would have approximated it a long time ago.

Would one of you be able to possibly hint where I'm going wrong, or which form you'd take?

**pickslides** · March 8th 2011, 10:39 PM

Here's a similar example for your consideration.

Areas of Surfaces of Revolution

**mr fantastic** · March 9th 2011, 02:11 AM

Originally Posted by Malaclypse

Been beating my head against the wall on an integral for awhile now....

The question is this:

Compute the surface area generated by revolving the curve $f(x) = 2\sqrt x ,\,\,{\text{for 1}} \leqslant x \leqslant 4$ about the x-axis.

I took the 1st derivative of the equation:

$f'(x) = \frac{1}{x}$

Easy enough. Then I went into my integral:

$2\pi \int_{\,1}^{\,4} {2\sqrt x \sqrt {1 + \frac{1}{{{x^2}}}} } dx$

That didn't look very pretty, and I didn't seem to be getting anywhere trying to solve it (no u-sub, and integration by parts didn't seem to be working), so I tried to manipulate the equation several times to see if a different variation would work...

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

and

$4\pi \int_{\,1}^{\,4} {\left( {\sqrt {\frac{1}{x}} } \right)} \sqrt {\left( {{x^2} + 1} \right)} dx$

and

$4\pi \int_{\,1}^{\,4} {\left( {\sqrt {\frac{1}{x}} } \right)} \sqrt {{x^2}\left( {1 + \frac{1}{{{x^2}}}} \right)} dx$

and

$4\pi \int_{\,1}^{\,4} {x\sqrt {\left( {\frac{{{x^2} + 1}}{{{x^3}}}} \right)} } dx$

But I can't find one to work out. My professor stated on the assignment that the integral can be done exactly, otherwise I would have approximated it a long time ago.

Would one of you be able to possibly hint where I'm going wrong, or which form you'd take?

Your calculation of f'(x) is wrong.

**HallsofIvy** · March 9th 2011, 03:06 AM

The derivative of $ln(x)$ is $\frac{1}{x}$ , not the derivative of $x^{1/2}$ . To differentiate that, use the power rule.

**Malaclypse** · March 9th 2011, 07:43 AM

Well now I feel like an idiot. Sure enough, I forgot to place the radical in the denominator when I performed it and didn't give it second thought. 1 / sqrt(x), NOT 1/x. I even looked at it several times. Urgh. Sorry all, and thanks.

That makes the integral nice and easy. What a waste of 3 hours.

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