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Lab07 Triple Integrals - volume
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Lab 07 - The "square-root solid"
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<p>Consider the volume integral over the solid $S$ given by
$$V=\iiint_S dV=\int_{-1}^1\int_{-\sqrt{1-x^2}}^{\sqrt{1-x^2}}
\int_{z=\sqrt{x^2+y^2}}^1\,dz\,dy\,dx$$

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<li>[<i>GeoGebra</i>] 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.  

<li>
Using geometry, what should the volume of this solid be? Justify your answer.


<li>[CoCalc for all of the following] Carry out the triple integral and see if you get the same answer as your geometry-based one.

<li>
Rewrite the volume integral as  $V=\iiint_S \,dz\,dx\,dy$ with appropriate limits.  (Put this in a Markdown cell).

<br><br>Then carry out the triple integral and calculate the value

<li >Rewrite the volume integral as  $V=\iiint_S \,dx\,dy\,dz$ with appropriate limits. (Markdown cell).

<br><br>Then carry out the triple integral and calculate the value






<li>[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.

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