Finding Volume of the solid, Integral

[lovemake1]

Homework Statement

Question Reads: A circular disk x ^2 + y^2 <= a ^ 2 , a > 0 is revolved about the line x = a.
Find the volume of the resulting solid.

Homework Equations

v = integral(a, b) (2pi)y [F(y) - G(y)] dy

The Attempt at a Solution

Im currently confused, should i take intergral with respect to x or y?
and what does this x = a tell me ?
does it just mean integral from (0, a)

 

[Mark44]

Homework Statement

Question Reads: A circular disk x ^2 + y^2 <= a ^ 2 , a > 0 is revolved about the line x = a.
Find the volume of the resulting solid.

Homework Equations

v = integral(a, b) (2pi)y [F(y) - G(y)] dy

The Attempt at a Solution

Im currently confused, should i take intergral with respect to x or y?

It depends on whether you use washers or cylindrical shells for your typical volume elements.

I'm guessing that you haven't drawn a sketch of the disk, and one of the solid of revolution. If that's the case, draw them. It's harder to get a handle on these kinds of problems if you don't have a good sense of what the region being revolved and the resulting solid look like.

and what does this x = a tell me ?
does it just mean integral from (0, a)

x = a is the vertical line that the disk (the circle and its interior) is revolved around.

 

[lovemake1]

ok so since this is a shell method.
i would have to represent in y integral.sqrt(y^2 - a ^2) = x
V= integral(0, a) 2pix(sqrt(y^2 - a ^2))dx

is that correct?

 

[Mark44]

ok so since this is a shell method.
i would have to represent in y integral.

I don't understand what you mean. If you mean an integral with dy, then no.

sqrt(y^2 - a ^2) = x

There's no reason to solve for x, but there's a very good reason to solve for y. In any case, your equation above is incorrect. If you square both sides, you get y² - a² = x², or y² - x² = a². That's not what you started with.

V= integral(0, a) 2pix(sqrt(y^2 - a ^2))dx

is that correct?

No. Did you draw a picture? If you had, you would see that the interval of integration is not [0, a].
In my drawing, this is the formula for the typical volume element. What is the interval over which $\Delta x$ ranges?
$$\Delta V = 2\pi (a - x)2y\Delta x$$