Integrals and The Washer Method

[erok81]

Homework Statement

Find the volume of the solid using The Washer Method

$$y=x^2, x=2$$

Homework Equations

The Attempt at a Solution

I can solve this problem fine and I don't think my actual question even affects the solution, but I would like to know how one does this.

In my textbook they say to setup the problem on a graph, no problem there. But I am confused on how one obtains the inner line segment.

Actually here is a drawing since I can't explain what I am looking for.

How do you find that red line segment? I would think that would matter in the solution, but it doesn't seem to - at least at my level.

 

[Mark44]

The length of the red line segment is 2 - sqrt(y), which is the difference of the larger x value (2) and the smaller x value (sqrt(y)).

You haven't described the region whose volume you want to find. Is the area you show to be revolved around the x-axis or the y-axis, or even some other line?

From the red line segment you show, it would seem that your region is being revolved around the y-axis.

 

[erok81]

Actually let me ask a real problem question. My prof failed to teach any of this to us, so I am having to learn this on my own.

Here is the problem.

The region in the first quadrant is bounded on the left by the circle $$x^2 + y^2 =3$$, on the right by the line x=sqrt(3), and above by the line y=sqrt(3).

I can draw the picture fine, but I don't understand to figure out R(x) and r(x) to solve. Any hints so I can get it setup?

Object is revolving about the y-axis.

 

[erok81]

The length of the red line segment is 2 - sqrt(y), which is the difference of the larger x value (2) and the smaller x value (sqrt(y)).

You haven't described the region whose volume you want to find. Is the area you show to be revolved around the x-axis or the y-axis, or even some other line?

From the red line segment you show, it would seem that your region is being revolved around the y-axis.

Got it. And yes it is revolving around the y-axis. I didn't know anything about the red line except it was placed there in my answer book. :)

 

[Mark44]

Each washer has a volume of pi(R(x)^2 - r(x)^2)*delta_y,
$$\Delta V = \pi[(R(x)^2 - r(x)^2]\Delta y$$
where R(x) = sqrt(3) and r(x) = sqrt(3 - y^2).



Your washers extend from y = 0 to y = sqrt(3).