Stumped on an integral.

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Mar 8th 2011, 10:11 PM
Malaclypse

Stumped on an integral.

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

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?
Mar 8th 2011, 10:39 PM
pickslides

Here's a similar example for your consideration.

Areas of Surfaces of Revolution
Mar 9th 2011, 02:11 AM
mr fantastic

Quote:

Originally Posted by Malaclypse

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

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.
Mar 9th 2011, 03:06 AM
HallsofIvy

The derivative of $ln(x)$ is $\frac{1}{x}$ , not the derivative of $x^{1/2}$ . To differentiate that, use the power rule.
Mar 9th 2011, 07:43 AM
Malaclypse

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.