Activity 12.6.3.
Let \(S\) be the solid cone bounded by \(z = \sqrt{x^2+y^2}\) and \(z=3\text{.}\) A picture of \(S\) is shown in Figure 12.6.4. Our goal in this activity is to set up an iterated integral of the form
\begin{equation}
\int_{x=?}^{x=?} \int_{y=?}^{y=?} \int_{z=?}^{z=?} \delta(x,y,z) \, dz \, dy \, dx\tag{12.6.1}
\end{equation}
to represent the mass of \(S\) in the setting where \(\delta(x,y,z)\) tells us the density of \(S\) at the point \((x,y,z)\text{.}\) Our particular task is to find the limits on each of the three integrals.
(a)
Remember that for the inner most integral of Equation (12.6.1), we will be looking for bounds on the \(z\)-coordinate for fixed values of \(x\) and \(y\text{.}\) In Figure 12.6.4, you can use the sliders to change the values of \(x\) and \(y\) and the plot will show the range of \(z\)-coordinates in our region for the particular values of \(x\) and \(y\text{.}\)
You should try several values of \(x\) and \(y\) and look at how the length of the segment changes in the \(z\)-direction. In particular, there is the same surface as the bottom boundary and the same surface as top boundary for every \(x\) and \(y\) pair. Give the equations for the top and bottom boundaries (in terms of \(x\) and \(y\))
\begin{align*}
z_{\text{top}}(x,y)\amp= \underline{\hspace{4cm}} \\
z_{\text{bottom}}(x,y)\amp= \underline{\hspace{4cm}}
\end{align*}
(b)
Now that we have our upper and lower boundaries for \(z\) as a function of a fixed choice of \(x\) and \(y\text{,}\) we need to look at what set of \((x,y)\) ,must be considered. Notice that if you choose a \((x,y)\) choice in Figure 12.6.4 that does not intersect our volume (like \((x,y)=(2.6,-2.3)\)), then the point shown is red. Write a couple of sentences to explain why the set of \((x,y)\) points we need to consider is NOT the square \([-3,3]\times[-3,3]\text{.}\)
(c)
On Figure 12.6.5, draw a plot of \(D\text{,}\) the region of \((x,y)\) points that you need to consider as part of \(S\text{.}\)
(d)
We now have part of our iterated integral
\begin{equation*}
\iiint_S \delta(x,y,z) dV = \int_D \left[ \int_{z_{\text{bottom}}(x,y)}^{z_{\text{top}}(x,y)} \delta(x,y,z) dz \right] dA
\end{equation*}
but we will need to give our region \(D\) from the previous task using a veritically simple description in order to have our iterated integral fit the form of Equation (12.6.1).
Give the inequalities that describe your region \(D\) from the previous task as vertically simple.
\begin{align*}
\underline{\hspace{4cm}} \amp\leq x \amp\leq \underline{\hspace{4cm}} \\
\underline{\hspace{4cm}} \amp\leq y \amp\leq \underline{\hspace{4cm}}
\end{align*}
(e)
In conclusion, write an iterated integral of the form (12.6.1) that represents the mass of the cone \(S\text{.}\)
