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When you are doing a triple integral and convert it to cylindrical co ordinates, how do you find the new ranges of integration?

I understand the new range of z, if z is between f(x,y) and g(x,y), you just sub in

x = r cos θ and y = r sin θ to find the new functions. But how do I find the range for r and θ? I'm lost there.

Here is a question i was working on and the solution is a bit confusing.

Find the volume of the solid bounded by the paraboloid z = 4x^2 +y^2 and the cylinder

y^2 + z = 2.

So the solution, they replace x = r cos θ / root 2 and y = r sin θ. I'm confused why not just use x = r cos θ? and the range for r is from 0 to 1 and for θ its between 0 and 2pi.

2. Relevant equations

3. The attempt at a solution

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# Homework Help: Triple integral & cylindrical coordinates

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