Lab 07 - The "square-root solid"

Consider the volume integral over the solid $S$ given by $$V=\iiint_S dV=\int_{-1}^1\int_{-\sqrt{1-x^2}}^{\sqrt{1-x^2}} \int_{z=\sqrt{x^2+y^2}}^1\,dz\,dy\,dx$$

1. [GeoGebra] Plot together the surfaces $z=u_1(x,y)=\sqrt{x^2+y^2}$ and $z=u_2(x,y)=1$ which are the lower and upper surfaces bounding the volume to be found.
2. Using geometry, what should the volume of this solid be? Justify your answer.
3. [CoCalc for all of the following] Carry out the triple integral and see if you get the same answer as your geometry-based one.
4. Rewrite the volume integral as $V=\iiint_S \,dz\,dx\,dy$ with appropriate limits. (Put this in a Markdown cell).
   
   Then carry out the triple integral and calculate the value
5. Rewrite the volume integral as $V=\iiint_S \,dx\,dy\,dz$ with appropriate limits. (Markdown cell).
   
   Then carry out the triple integral and calculate the value
6. [To finish Wednesday] Referring back to the integral at the beginning of this assignment: Evaluate the $dz$ integral, then convert the remaining double-integral to polar coordinates, $(r,\theta)$. Write down (below) your double integral in polar coordinates and then evaluate the result to show (hopefully!) that this also gives the same answer.