[Solved] Calculus: Finding Volume of Solids by Shells

[dominet]

Hello. My homework requires me to find the volume of a solid by shells but my Math teacher only explained a simple exam with only 1 function revolved about the x-axis or y-axis where the radius is either just "y" or "x".

In my homework, I see questions like this:

Find Volume by shell method of region bounded by: y=6x+7 and y=x^2

about the line: a) x= -1 b) x= 7

 

I don't know how to find the radius here.

My attempt at "a)" was: integral of 2pi*((-1-x)(6x+7-x^2))*dx from -1 to 7

but that gave me a negative answer.

 

Can someone explain to me how to usually find the radius and height of the shell for questions of this form?

Thank you.

Edited October 21, 2011 by dominet

[Schrödinger's hat]

Hello. My homework requires me to find the volume of a solid by shells but my Math teacher only explained a simple exam with only 1 function revolved about the x-axis or y-axis where the radius is either just "y" or "x".

In my homework, I see questions like this:

Find Volume by shell method of region bounded by: y=6x+7 and y=x^2

about the line: a) x= -1 b) x= 7

 

I don't know how to find the radius here.

My attempt at "a)" was: integral of 2pi*((-1-x)(6x+7-x^2))*dx from -1 to 7

but that gave me a negative answer.

 

Can someone explain to me how to usually find the radius and height of the shell for questions of this form?

Thank you.

 

Could you explain your reasoning a little more? I think that perhaps noone has answered because they're not sure what you did. If you give us a little more context we're more likely to be able to find your mistake.

 

I'd approach this by changing to cylindrical coordinates and doing the whole integral from scratch, but that's largely because calc 1 was a while ago and I don't remember the shortcut for solids of revolution. I can explain this if you like, but chances are high that that's not the method that was intended for solving this problem.

 

 

Use of the forum's latex support is appreciated to make reading it easier

Something along the lines of

[math]V=\int\limits_{\pi}^{e}\int\limits_a^b f_{\text{foo}}\times x^3\;dxdy[/math]

To produce:

[math]V=\int\limits_{\pi}^{e}\int\limits_a^b f_{\text{foo}}\times x^3\;dxdy[/math]

[dominet]

I realized the problem was just that I should have used absolute value for the radius.

 

So the answer would be:

 

integral of 2pi*(|-1-x|(6x+7-x^2)).dx from -1 to 7