There are a region bounded by the curve y=cos^2(x)sin^4(x) and the x-axis , where pi<x<2pi. Find the volumn of the solid of revolution when the region is revolved about the y-axis
Hey Lau, I solved this problem using the shell method.
It is best to start with a rough sketch of the function on the given interval (Will give a M shape), and once you do you will realize that we should split the region again this time considering: pi <x<3pi/2 and 3pi/2<x<2pi. This is because from x=pi to x=3pi/2 , the function "starts" then "ends". Also, this will help us in the calculations as the function "repeats" its self on 3pi/2 < x < 2pi.
Using the shell method, I found the height of a shell to be: $\displaystyle h=f(x)= \sin^4x \cos^2x$ and a radius about the y axis to be $\displaystyle r=x$.
Then the volume of the first cycle of the periodic function is:
$\displaystyle V= \int_{\pi}^{3 \pi/2} 2 \pi r \cdot h \, dx$
$\displaystyle V= \int_{\pi}^{3 \pi/2} 2 \pi (x) \cdot (\sin^4x \cos^2x) \, dx = \frac{\pi}{36} + \frac{5 \pi}{64} $
So finally, the total volume is just $\displaystyle 2V = \frac{\pi}{18} + \frac{5 \pi}{32} \approx 5.02$
@sakonpure6, you have a conceptual mistake.
$\displaystyle 3.30405 \approx \int_{1.5\pi }^{2\pi } {\left( {2\pi x} \right)\left( {{{\sin }^4}(x){{\cos }^2}(x)} \right)dx} \ne \int_\pi ^{1.5\pi } {\left( {2\pi x} \right)\left( {{{\sin }^4}(x){{\cos }^2}(x)} \right)dx \approx 2.0963} $
Look at the graph again.
Oh thanks for pointing that out Plato! So, Lau the correct answer is: 3.304 +2.096 = 5.40
Edit: Plato, did something in the equation give away that the function was not periodic - had different amplitudes? Or did you plot to verify?
It is best to start with a rough sketch of the function on the given interval (Will give a M shape), and once you do you will realize that we should split the region again this time considering: pi <x<3pi/2 and 3pi/2<x<2pi. This is because from x=pi to x=3pi/2 , the function "starts" then "ends". Also, this will help us in the calculations as the function "repeats" its self on 3pi/2 < x < 2pi.
This is wrong. In fact, it completely misses the point of this whole thread!
@dungtuyet, did you even read the entire thread? If you did, what part did you not get?
One more time. Here is a standard example.
What is the volume of the graph of $y=|x-3|,~ 2\le x\le 4$ about the $y \text{-axis}~?$
Please study each of these: from $x=2\to 3$; then from $x=3\to 4$
Note that those two volumes are not equal even though the graph is symmetric about $x=3$. The radii are different?
Add those two values and compare the result with this.