Finding Volume of E Bounded by x^2+z^2-y^2=2, y=1, and y=7

[quietrain]

Homework Statement

find volume E bounded by x²+z²-y²=2 , planes y=1 and y=7

The Attempt at a Solution

if i do it this way,

triple integral 1 dydA

may i know what are my limits for dy? is it just 1 to 7? or 1 to sqrt(x²+z²-2) ?and also, why is it that the double integral of z da also gives the volume? are they the same? can i use them for any questions?

thanks!

 

[LCKurtz]

Homework Statement

find volume E bounded by x²+z²-y²=2 , planes y=1 and y=7

The Attempt at a Solution

if i do it this way,

triple integral 1 dydA

may i know what are my limits for dy? is it just 1 to 7? or 1 to sqrt(x²+z²-2) ?

Neither. And integrating dy first is a very poor choice for this problem.

and also, why is it that the double integral of z da also gives the volume? are they the same? can i use them for any questions?

thanks!

Have you drawn a graph of this figure? That should always be your first step. Once you do that you may see that the easiest way to do this problem is to use the fact that it is a volume of revolution about the y axis.

Judging from your question, my suggestion at this point is for you to make an appointment with your teacher and go over this material. There are too many issues to type about.

 

[quietrain]

ouch... volume of revolution? :(

is drawing the diagram the only way?

because what if i can't draw it out? then i have no way of doing?

my notes says there are generally 3 types , where the projection is on the x-y , x-z and y-z planes respectively.

in this case i can't project it on the x-z plane? since my y effectively cuts of the planes at the sides? anyway, just another question, why is the equation of a cylinder x2 + y2 = 1 ? it doesn't even involve the z coordinate? issn't that just a circle?

 

[LCKurtz]

ouch... volume of revolution? :(

Time to get out your Calc I or II book and review.

is drawing the diagram the only way?

If you want to have any confidence in getting correct limits, yes. Otherwise you are just making an educated guess which is likely to be incorrect as in this problem.

because what if i can't draw it out? then i have no way of doing?

my notes says there are generally 3 types , where the projection is on the x-y , x-z and y-z planes respectively.

That's true. The problem is in deciding to dx, dy, or dz direction first. You can always do any order you wish, but some choices are more difficult than others. That is why a picture is useful; it can help you make a good choice.

in this case i can't project it on the x-z plane? since my y effectively cuts of the planes at the sides?

Yes, you could project on the xz plane, meaning do the dy integral first. But if you look at the graph, you will see you have to break the integral into two parts making twice as much work or, worse, getting the problem wrong because you don't see the difficulty with no picture.

anyway, just another question, why is the equation of a cylinder x2 + y2 = 1 ? it doesn't even involve the z coordinate? issn't that just a circle?

Use the X² icon for superscripts. Yes, that is a circle in the xy plane. But it is also a circle in the plane z = 1 or 2 or anything so you get a cylindrical surface. This is typical of equations of surfaces with one variable missing.

 

[quietrain]

oh gosh... so this is a hyperboloid of 1 sheet with symmetry in the y-axis?

oh i see, so the volume is probably split half in positive planes, half in negative planes, with the y-axis running through the origin? so if i integrate the dy first, i would just have to times 2 to the answer?

so my limits of x will be halved, while my z remains?