Multiple integrals for finding volume

[aruwin]

How do I solve this? How do I determine the range? Ill they be triple integrals?Please explain to me.

Find the volumes in R³.

1. Find the volume U that is bounded by the cylinder surface x^2+y^2=1 and the plane
surfaces z=2, x+z=1.

2. Find the volume W that is bounded by the cylindrical surfaces
x^2 + y^2 = 1 and x^2 + z^2 = 1.

I know these are 2 questions and they're both about finding volumes but the second one seems to have 2 cylinders ??I don't get it.

 

[tiny-tim]

hi aruwin!

these integration problems are basically about slicing the volume in a convenient way

for the first one …

1. Find the volume U that is bounded by the cylinder surface x^2+y^2=1 and the plane
surfaces z=2, x+z=1.

… horizontal slices seem sensible:

what is the shape of the horizontal slice at a general height z ?

2. Find the volume W that is bounded by the cylindrical surfaces
x^2 + y^2 = 1 and x^2 + z^2 = 1.

I know these are 2 questions and they're both about finding volumes but the second one seems to have 2 cylinders ??I don't get it.

yes, two cylinders

again, try horizontal slices, write the second condition as x² = 1 - z², and remember the RHS is a constant (for that slice)

 

[aruwin]

hi aruwin!

these integration problems are basically about slicing the volume in a convenient way

for the first one …


… horizontal slices seem sensible:

what is the shape of the horizontal slice at a general height z ?


yes, two cylinders

again, try horizontal slices, write the second condition as x² = 1 - z², and remember the RHS is a constant (for that slice)

Horizontal slice?You mean, like a triangle?? And what do you mean by RHS is a constant?

 

[tiny-tim]

Horizontal slice?You mean, like a triangle??

won't one side be curved?

And what do you mean by RHS is a constant?

in x² = 1 - z², z is constant for any particular slice

 

[aruwin]

won't one side be curved?


in x² = 1 - z², z is constant for any particular slice

LOL,of course one side should be a curve. My bad, I meant horizontal slice as in a slanting plane. Is that what you refer to as horizontal slice?

 

[tiny-tim]

LOL,of course one side should be a curve. My bad, I meant horizontal slice as in a slanting plane. Is that what you refer to as horizontal slice?

no, a horizontal slice means between the horizontal planes at heights z and z+dz

 

[aruwin]

hi aruwin!

these integration problems are basically about slicing the volume in a convenient way

for the first one …


… horizontal slices seem sensible:

what is the shape of the horizontal slice at a general height z ?


yes, two cylinders

again, try horizontal slices, write the second condition as x² = 1 - z², and remember the RHS is a constant (for that slice)

Can you show me a diagram for the first cylinder?I am guessing it looks like a cylinder with the one slanting surface but I can't picture the XYZ coordinate inside it. I really need to see a picture.

 

[tiny-tim]

I am guessing it looks like a cylinder with the one slanting surface but I can't picture the XYZ coordinate inside it.

yes, it is a cylinder with one slanting surface

but the easiest way of integrating is to divide it into horizontal slices (so each slice is a "damaged" circle with a bit sliced off)

you could integrate by dividing it into slanting slices, but

i] you'd have to define a new parameter x+z, and integrate with respect to that

ii] you'd need to know the formula for the area of an ellipse (ok, i admit that's fairly easy)

iii] you'd still have a problem with "damaged" circles at the bottom!​

 

[haruspex]

1. Find the volume U that is bounded by the cylinder surface x^2+y^2=1 and the plane surfaces z=2, x+z=1.

Before leaping into calculus, think what this shape looks like. What shape might you make with two of them?

two cylinders

again, try horizontal slices, write the second condition as x² = 1 - z², and remember the RHS is a constant (for that slice)

I think this one would be much easier using vertical slices in the x-z plane. The slices would be squares.

 

[tiny-tim]

I think this one would be much easier using vertical slices in the x-z plane. The slices would be squares.

oooh yes … horizontal slices work, but vertical slices are easier!