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The original problem is:

"Find the volume of the solid obtained by rotating about the x axis the region enclosed by the curves $y = \frac{9}{x^2 + 9},y=0,x=0,\,$and $x = 3$"

I set up the following integral $$81\pi\int_0^3\frac{1}{(x^2 + 9)^2}dx$$ using the cylinder method (I believe it's called like that) and when I calculated it using a computer I obtained the correct answer but I have been having difficulties in solving it manually. I tried the shell method as well and I didn't see it any easier to solve but I may be wrong of course.

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2 Answers

up vote 2 down vote accepted

We want to find $$\int \frac{dx}{(x^2+9)^2}.$$ Let $x=3u$. Apart from a constant factor, we end up with $$\int\frac{du}{(u^2+1)^2}.$$

Let $u=\tan t$ (we could have gone more directly, by letting $x=3\tan t$). Then $du=\sec^2 t\,dt$, and we end up with $$\int \frac{\sec^2 t}{\sec^4 t}\,dt.$$ This is the familiar integral $$\int \cos^2 t\,dt.$$ A common approach to this is to use the fact that $\cos 2t=2\cos^2 t-1$.

e can save time if we make the substitutions on the definite integral, that is, substitute for the endpoints.

Another way: Consider $\displaystyle\int \frac{dx}{x^2+9}$. Attack this by integration by parts, letting $du=dx$ and $v=\frac{1}{9+x^2}$. Then we can take $u=x$. Also, $dv=\frac{-2x\,dx}{(x^2+9)^2}$. Thus $$\int \frac{dx}{x^2+9}=\frac{x}{x^2+9}+\int\frac{2x^2\,dx}{(x^2+9)^2}.$$ But $2x^2=2x^2+18-18$. Thus $$\int\frac{dx}{x^2+9}=\frac{x}{x^2+9}+2\int\frac{dx}{x^2+9}-18\int\frac{dx}{(9+x^2)^2}.$$ We will be finished if we can find $\displaystyle\int\frac{dx}{x^2+9}$. The substitution $x=3w$ turns this into a familiar integral.

Remark: In the second approach, we obtained a reduction formula. A very similar trick expresses $\displaystyle\int \frac{dx}{(x^2+9)^{n+1}}$ in terms of $\displaystyle\int \frac{dx}{(x^2+9)^{n}}$

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Do you suggest the students memorize a suitable formula for the integral the OP faced above? Or you always prefer to apply methods to find the best approach? Thanks. + –  B. S. Jan 4 '13 at 8:16
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I have always suggested to students that they remember approaches. Many prefer to try to memorize formulas. In the short run such memorization is often ineffective. In the long run it is disastrous. –  André Nicolas Jan 4 '13 at 8:24
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Although the integral you've got can be evaluated by some technical method, as @Andre noted completely; you can use The cylinder method instead. According to this method we consider the following integral: $$V=2\pi\int_{y=a}^{y=b}x|f(x)|dx$$ So we have here: $$\int_0^118\pi x\frac{1}{x^2+9}dx$$. enter image description here

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+1 Nice picture! –  Mario Carneiro Jan 4 '13 at 7:59
    
@MarioCarneiro: Thanks for your support. –  B. S. Jan 4 '13 at 8:13
    
+1 phenomenal animation/graphic! –  amWhy Feb 25 '13 at 0:05
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