MHB Henry's question at Yahoo Answers concerning a surface of revolution

MarkFL
Gold Member
MHB
Messages
13,284
Reaction score
12
Here is the question:

Calculus 2 help me please?

Find the surface area generated by rotating the given curve about the y-axis.

x = 3t^2, y = 2t^3, 0 ≤ t ≤ 5

Here is a link to the question:

Calculus 2 help me please? - Yahoo! Answers

I have posted a link there to this topic so the OP can find my response.
 
Mathematics news on Phys.org
Hello Henry,

We are asked to find the surface of rotation of the curve described parametrically by:

$x(t)=3t^2$

$y(t)=2t^3$

with $t$ in $[0,5]$.

Since the axis of rotation is the $y$-axis and $x(t)$ is non-negative on the given interval for $t$, we may use:

$\displaystyle S=2\pi\int_0^5 x(t)\sqrt{\left(\frac{dx}{dt} \right)^2+\left(\frac{dy}{dt} \right)^2}\,dt$

So, we compute:

$\displaystyle \frac{dx}{dt}=6t$

$\displaystyle \frac{dy}{dt}=6t^2$

and we have:

$\displaystyle S=2\pi\int_0^5 3t^2\sqrt{\left(6t \right)^2+\left(6t^2 \right)^2}\,dt$

$\displaystyle S=36\pi\int_0^5 t^3\sqrt{1+t^2}\,dt$

Now, let's use the substitution:

$\displaystyle t=\tan(\theta)\,\therefore\,dt=\sec^2( \theta)\,d \theta$ and we have:

$\displaystyle S=36\pi\int_0^{\tan^{-1}(5)} \tan^3(\theta)\sqrt{1+\tan^2(\theta)}\,sec^2( \theta)\,d \theta$

$\displaystyle S=36\pi\int_0^{\tan^{-1}(5)} \tan^3(\theta)\sec^3(\theta)\,d\theta$

$\displaystyle S=36\pi\int_0^{\tan^{-1}(5)} \frac{\sin(\theta)(1-\cos^2(\theta))}{\cos^6(\theta)}\,d\theta$

Now, let's try the substitution:

$u=\cos(\theta)\,\therefore\,du=-\sin(\theta)\,d \theta)$ and we have:

$\displaystyle S=36\pi\int_{\frac{1}{\sqrt{26}}}^{1} \frac{1-u^2}{u^6}\,du$

$\displaystyle S=36\pi\int_{\frac{1}{\sqrt{26}}}^{1} u^{-6}-u^{-4}\,du$

$\displaystyle S=36\pi\left[\frac{u^{-5}}{-5}-\frac{u^{-3}}{-3} \right]_{\frac{1}{\sqrt{26}}}^{1}$

$\displaystyle S=\frac{12\pi}{5}\left[5u^{-3}-3u^{-5} \right]_{\frac{1}{\sqrt{26}}}^{1}$

$\displaystyle S=\frac{12\pi}{5}\left(\left(5(1)^{-3}-3(1)^{-5} \right)- \left(5\left(\frac{1}{\sqrt{26}} \right)^{-3}-3\left(\frac{1}{\sqrt{26}} \right)^{-5} \right) \right)$

$\displaystyle S=\frac{12\pi}{5}\left(5-3-130\sqrt{26}+2028\sqrt{26} \right)$

$\displaystyle S=\frac{12\pi}{5}\left(2+1898\sqrt{26} \right)$

$\displaystyle S=\frac{24\pi}{5}\left(1+949\sqrt{26} \right)$
 
Just a remark to reduce computations ..

Here we don't need a geometric substitution :S=36\pi\int_0^5 t^3\sqrt{1+t^2}\,dt

rewrite as S=18\pi\int_0^5 2t \cdot t^2\sqrt{1+t^2}\,dtWe can use the substitution u=1+t^2 \,\,\Rightarrow \,\, t^2=u-1so we have du=2t\, dtThe integral becomes as the following :S=18\pi \int_1^{26} (u-1)\sqrt{u}\,duS=18\pi \int_1^{26} \sqrt{u^3}-\sqrt{u}\,du
 
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation $$ a^n+b^n=c^n $$ has no solutions with positive integers if ##n>2.## It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book Arithmetica by Diophantus of Alexandria. It gained popularity because Fermat noted in his copy "Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et...
I'm interested to know whether the equation $$1 = 2 - \frac{1}{2 - \frac{1}{2 - \cdots}}$$ is true or not. It can be shown easily that if the continued fraction converges, it cannot converge to anything else than 1. It seems that if the continued fraction converges, the convergence is very slow. The apparent slowness of the convergence makes it difficult to estimate the presence of true convergence numerically. At the moment I don't know whether this converges or not.
Back
Top