Integrating the Region of a Ball Cut by a Cylinder | Finding Limits for x, y, z

[jonroberts74]

Homework Statement

I have to describe by finding the limits of integrations for x,y,z

the region of the ball
$$x^2 + y^2 +z^2 \le 4$$ cut by $$2x^2+z^2=1$$

The Attempt at a Solution

so I can visualize these without much trouble and I used grapher so I have a working model.

I also put in the y=0 plane because I figure "slicing" the cylinder with y_{0} is the way to go.

so $$-\sqrt{4-x^2-z^2} \le y \le \sqrt{4-x^2-z^2}$$

then I can look at the y=0 plane to see the 2-D ellipse, here's a link

http://www.wolframalpha.com/input/?i=plot+x^2+++z^2/2+=1/2

now if I "slice" vertically $$-1 \le z \le 1$$ and find the change of x

so $$-\sqrt{\frac{1-z^2}{2}} \le x \le \sqrt{\frac{1-z^2}{2}}$$

is that correct?

 

[slider142]

The six inequalities you derived do indeed correctly describe the region of integration.

 

[LCKurtz]

I agree with slider42 that your limits are correct, but I wouldn't consider the problem finished until you set up a triple integral (assuming you are calculating a volume) showing the order of integration with proper limits on each integral.

 

[jonroberts74]

yes I agree, question only asked for that but more practice is always better. I'll come back to this problem