Respuesta :
Answer:
Required volume is [tex]\frac{4864\pi}{3}[/tex] unit.
Step-by-step explanation:
Given equations of curves,
[tex]y=32-x^2[/tex], [tex]y=x^2[/tex]
substitute second in first we will get,
[tex]x^2=36-x^2\implies x^2=16\implies x=\pm 4[/tex]
When [tex]x=4, y=16[/tex] and [tex]x=-4, y=16[/tex]. Thus both curves intersect at the points (4,16),(-4,16). And thus [tex]-4<x<4[/tex].
Considering width of the representative rectangle as [tex]\Delta x[/tex] which is parallel to x-axis because of considering cylindrical shell. Therefore volume of the shell is given by,
[tex]V_{Shell}=(\textit{Length of box})\times (\textit{Width of box})\times (\textit{Thikness of box})[/tex]
Now,
Length of generating box=Length of [tex]\Delta x[/tex] from line x=4(axis of revolution)=circumference of the shell=(4-x)
Width of box=[tex](36-x^2)-x^2[/tex]
Hence,
[tex]V_{Shell}[/tex]
[tex]=\int_{-4}^{4}2\pi (4-x)(36-2x^2)dx[/tex]
[tex]=2\pi\int_{-4}^{4}(144-36x-8x^2+2x^3)dx[/tex]
[tex]=2\pi\{144\big[x\big]_{-4}^{4}-18\big[x^2\big]_{-4}^{4}-\frac{8}{3}\big[x^3\big]_{-4}^{4}+\frac{1}{2}\big[x^4\big]_{-4}^{4}\}[/tex]
[tex]=\frac{4864\pi}{3}[/tex]
which is required volume of generating area.
