HFinding Z-Limits in a Solid Horn Rotated Around the Y-Axis

[jeff1evesque]

Homework Statement

Solid horn obtained by rotating the points $${[x=0], [0 \leq y \leq 4], [0 \leqz \leq \frac{1}{8}y^{2}] }$$ circles around y-axis of radius $$\frac{1}{8}y^2$$. Set up the integral dzdxdy.

Homework Equations

Cartesian coordinates.

The Attempt at a Solution

I don't understand how the z-limits are $$\pm \sqrt{\frac{y^4}{64} - x^2}$$? I understand that the z limits must involve x and y, but cannot come up with the latter conclusion.

Thanks

 

[Ja4Coltrane]

I think you have some typos. It's not clear what you mean here. You have, for instance, 0 is less than or equal to y^2/8. That gives no information as y^2 is greater than or equal to zero for all real numbers y.

 

[jeff1evesque]

I think you have some typos. It's not clear what you mean here. You have, for instance, 0 is less than or equal to y^2/8. That gives no information as y^2 is greater than or equal to zero for all real numbers y.

Well I suppose without looking at a picture, that may be a reasonable opinion- and hard to interpret. I've attached a picture, if that helps any.

Thanks a lot

JL

 

[jeff1evesque]

I guess I am having difficulty determining the limits of this particular integration (in cartesian) along the z-axis.

 

[Mark44]

It sometimes takes several hours for an attachment file to be approved, so if you could describe the solid in words, that would be helpful.

My guess as to how you have described the solid so far is that the curve in the x-y plane, x = y²/8, is revolved around the y-axis to form a solid. And you want the portion of this solid between the planes y = 0 and y = 4.

Is this a reasonable description?

 

[jeff1evesque]

It sometimes takes several hours for an attachment file to be approved, so if you could describe the solid in words, that would be helpful.

My guess as to how you have described the solid so far is that the curve in the x-y plane, x = y²/8, is revolved around the y-axis to form a solid. And you want the portion of this solid between the planes y = 0 and y = 4.

Is this a reasonable description?

Yup that sounds reasonable. Basically, if you could picture then end of a horn [perhaps a trumpet, beginning as a point on the origin and expanding out along the y-axis] with the y-axis going through the center, that's what this image looks like. At y = 4, the "horn" has a height of z = 2, which obviously rotates around the y-axis.

 

[jeff1evesque]

I actually solved this problem this morning with some help from other.

Thanks,


JL