Can you grunt it out

BigGlenntheHeavy

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Find volume of the solid formed by revolving the region bounded by the graphs of\displaystyle Find \ volume \ of \ the \ solid \ formed \ by \ revolving \ the \ region \ bounded \ by \ the \ graphs \ of

y = x3+x+1, y = 1, and x = 1 about the line x = 2, using the disc method, not shells.\displaystyle y \ = \ x^{3}+x+1, \ y \ = \ 1, \ and \ x \ = \ 1 \ about \ the \ line \ x \ = \ 2, \ using \ the \ disc \ method, \ not \ shells.

Note, youll have to solve y = x3+x+1 for x.\displaystyle Note, \ you'll \ have \ to \ solve \ y \ = \ x^{3}+x+1 \ for \ x.
 
OK, f(x) = y = x3+x+1, hence f(x) has one real negative solution and two imaginary ones.\displaystyle OK, \ f(x) \ = \ y \ = \ x^{3}+x+1, \ hence \ f(x) \ has \ one \ real \ negative \ solution \ and \ two \ imaginary \ ones.

For f(x), we get x= (y12+4+27(1y)2108)1/3+(y124+27(1y)2108)1/3\displaystyle For \ f(x), \ we \ get \ x= \ \bigg(\frac{y-1}{2}+\sqrt\frac{4+27(1-y)^{2}}{108}\bigg)^{1/3}+\bigg(\frac{y-1}{2}-\sqrt\frac{4+27(1-y)^{2}}{108}\bigg)^{1/3}

[Washer Method]\displaystyle [Washer \ Method]

V=π13[([((y12+4+27(1y)2108)1/3+(y124+27(1y)2108)1/3)2]2)1]dy=\displaystyle V=\pi\int_{1}^{3}\bigg[\bigg(\bigg[\bigg(\bigg(\frac{y-1}{2}+\sqrt\frac{4+27(1-y)^{2}}{108}\bigg)^{1/3}+\bigg(\frac{y-1}{2}-\sqrt\frac{4+27(1-y)^{2}}{108}\bigg)^{1/3}\bigg)-2\bigg]^{2}\bigg)-1\bigg]dy=

29π15\displaystyle \frac{29\pi}{15}

[Shell Method]\displaystyle [Shell \ Method]

Definitely Shell Method\displaystyle Definitely \ Shell \ Method

V=2π01(2xx2+2x3x4)dx = 29π15\displaystyle V=2\pi\int_{0}^{1}(2x-x^{2}+2x^{3}-x^{4})dx \ = \ \frac{29\pi}{15}

For further elucidation, see graph below.\displaystyle For \ further \ elucidation, \ see \ graph \ below.

[attachment=0:3s7482px]abc.jpg[/attachment:3s7482px]
 

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Cool, Glenn. Is that Cardano's method for solving a cubic?. I have not seen, nor used, that in years
 
It was one of the pre-renaissance Italians, I think there were two - Cardano was one of them.

With today's technology, there isn't any need to solve cubics algebraically, as one's trusty TI-89 will give you all and any solution(s); just killing time in my "Golden Years".
 
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