1. ## volume of revolution

i have done the part a, for b , i use the key in the (circled part equation ) in to calculator .. my ans is also different form the ans given. is my concept correct by the way?

2. ## Re: volume of revolution

I haven't see in your calculations to multiply by π .
I found after integration from 0 to 1/2 V = 5.0477...
check it and do not forget to multiply by π.

3. ## Re: volume of revolution

ok, noted my mistake

4. ## Re: volume of revolution

i redo the question, now got stuck here... which part is wrong?

5. ## Re: volume of revolution

For one thing, a volume by revolution is given by \displaystyle \begin{align*} V = \pi \int_a^b{\left[ f(x) \right] ^2 \, dx} \end{align*}. You still don't have \displaystyle \begin{align*} \pi \end{align*}...

6. ## Re: volume of revolution

using calcultor, before times pi, the ans should be 1.60675, by my ans only 1.5... which part is wrong? this almost drive me crazy!

7. ## Re: volume of revolution

\displaystyle \begin{align*} V &= \pi \int_0^{\frac{1}{2}}{ \left( 1 + \frac{1}{4x^2 + 1} \right) ^2 \, dx } \\ &= \pi \int_0^{\frac{\pi}{4}}{ \left\{ 1 + \frac{1}{4 \left[ \frac{1}{2}\tan{(\theta)} \right] ^2 + 1 } \right\} ^2 \, \frac{1}{2}\sec^2{(\theta)}\,d\theta } \textrm{ after making the substitution } 2x = \tan{(\theta)} \\ &= \frac{\pi}{2}\int_0^{\frac{\pi}{4}}{ \left[ 1 + \frac{1}{\tan^2{(\theta)} + 1 } \right] ^2 \sec^2{(\theta)}\,d\theta } \\ &= \frac{\pi}{2}\int_0^{\frac{\pi}{4}}{ \left[ 1 + \frac{1}{\sec^2{(\theta)} } \right] ^2 \sec^2{(\theta)}\,d\theta} \\ &= \frac{\pi}{2} \int_0^{\frac{\pi}{4}}{ \left[ 1 + \cos^2{(\theta)} \right] ^2 \sec^2{(\theta)} \,d\theta } \\ &= \frac{\pi}{2} \int_0^{\frac{\pi}{4}}{ \left[ 1 + 2\cos^2{(\theta)} + \cos^4{(\theta)} \right] \sec^2{(\theta)}\,d\theta } \\ &= \frac{\pi}{2} \int_0^{\frac{\pi}{4}}{ \sec^2{(\theta)} + 2 + \cos^2{(\theta)} \,d\theta } \\ &= \int_0^{\frac{\pi}{4}}{ \sec^2{(\theta)} + 2 + \frac{1}{2} + \frac{1}{2}\cos{(2\theta)} \,d\theta } \\ &= \frac{\pi}{2} \int_0^{\frac{\pi}{4}}{ \sec^2{(\theta)} + \frac{5}{2} + \frac{1}{2}\cos{(2\theta)} \,d\theta } \\ &= \frac{\pi}{2} \left[ \tan{(\theta)} + \frac{5}{2}\theta + \frac{1}{4}\sin{(2\theta)} \right]_0^{\frac{\pi}{4}} \\ &= \frac{\pi}{2} \left\{ \left[ \tan{ \left( \frac{\pi}{4} \right) } + \frac{5}{2}\left( \frac{\pi}{4} \right) + \frac{1}{4}\sin{ \left( \frac{\pi}{2} \right) } \right] - \left[ \tan{(0)} + \frac{5}{2} \left( 0 \right) + \frac{1}{4}\sin{(0)} \right] \right\} \\ &= \frac{\pi}{2} \left[ \left( 1 + \frac{5\pi}{8} + \frac{1}{4} \right) - \left( 0 + 0 + 0 \right) \right] \\ &= \frac{\pi}{2} \left( \frac{5}{4} + \frac{5\pi}{8} \right) \\ &= \frac{5\pi}{8} \left( 1 + \frac{\pi}{2} \right) \\ &= \frac{5\pi}{16} \left( 2 + \pi \right) \end{align*}

8. ## Re: volume of revolution

thank you very much... i should have done in this way instead of expand the y^2

9. ## Re: volume of revolution

Originally Posted by delso
thank you very much... i should have done in this way instead of expand the y^2
Expanding first should also have worked though...