Finding the volume of a solid.

[tarmon.gaidon]

Homework Statement

Find the volume of the solid in the first octant bounded by the coordinate planes and the plane 2x+y-4=0 and 8x-4z=0.


This is a problem for a practice exam for my calculus course and I just need some help getting started.

I have had a lot of trouble in this course trying to figure what the bounds of my integration should be so any pointers would be appreciated!

P.S. I have also had a lot of trouble reversing the oder of integration and changing to spherical and cylindrical coordinates. Mainly because I have trouble figuring out how to change the bounds.

 

[Mark44]

Homework Statement

Find the volume of the solid in the first octant bounded by the coordinate planes and the plane 2x+y-4=0 and 8x-4z=0.


This is a problem for a practice exam for my calculus course and I just need some help getting started.

I have had a lot of trouble in this course trying to figure what the bounds of my integration should be so any pointers would be appreciated!

This is not surprising, since for many problems of this kind, finding the bounds of integration is the hardest part.

Have you drawn a picture of the solid? Drawing a picture should give you a good idea of what the region of integration looks like, and should help you get the limits of integration.

P.S. I have also had a lot of trouble reversing the oder of integration and changing to spherical and cylindrical coordinates. Mainly because I have trouble figuring out how to change the bounds.

 

[lanedance]

i wouldn't be using spherical or cylindrical, but would have a think about the volume - and try and draw it... what's you attempts at you bounds?

 

[tarmon.gaidon]

Hey Mark,

Thanks for the suggestion, I see what you are saying but let me ask this.

I have a problem here where I needed to change a triple integral from cylindrical coordinates to Cartesian. I have attached an image of the problem and the solution. When I went to solve it I sketched the solid and then attempted to write the bounds of each variable. I had come to the conclusion that y should be from $$\sqrt{1-x^2}$$ to 1 which is close but not quite right. How would I come to the conclusion they made instead of what I did?

 

[The Chaz]

Wow, that image would not help me learn ANYTHING!
The limits for "r" and theta will help you determine the limits for x and y.
Since theta only ranges from 0 to π and r ranges from 0 to 1, it is a SEMIcircle of radius one (the top half, actually). To integrate the top of the unit circle "dydx"...

The upper limit for a vertical representative rectangle will range from y = 0 (the x-axis) to the curve y = +√(1-x^2). (...I included the "+" to emphasize that it's the top half).

Then x ranges from -1 to 1.