17.7.07 other orders usually work well and are occasionally easier to evaluate

• MHB
• karush
In summary, the order of integration for cylindricsl coordinates is usually preferred, but other orders can work well and are occasionally easier to evaluate.
karush
Gold Member
MHB
The integrals we have seen so far suggest that there are preferred orders of integration for cylindricsl coordinates, but other orders usually work well and are occasionally easier to evaluate. Evaluate the integral

\begin{align*}\displaystyle
dV&=\int_{0}^{2\pi}\int_{0}^{3}\int_{0}^{z/3}r^3 \, dr \, dz \, d\theta\\
\\
&=\color{red}{\frac{3\pi}{10}}
\end{align*}

ok I tried some rearrange but it just got worse
I would presume this is converting $r^3$ to rectangular coordinates

karush said:
The integrals we have seen so far suggest that there are preferred orders of integration for cylindricsl coordinates, but other orders usually work well and are occasionally easier to evaluate. Evaluate the integral

\begin{align*}\displaystyle
dV&=\int_{0}^{2\pi}\int_{0}^{3}\int_{0}^{z/3}r^3 \, dr \, dz \, d\theta\\
\\
&=\color{red}{\frac{3\pi}{10}}
\end{align*}

ok I tried some rearrange but it just got worse
I would presume this is converting $r^3$ to rectangular coordinates
First, because neither the integrand nor the limits of the other two integrals involve $\theta$ we can do that separately:
$\int_0^{2\pi} d\theta= 2\pi$
so that is simply
$dV= 2\pi \int_0^3 \int_0^{z/3} r^3 dr dz$
The integral of $r^3$ is $\frac{r^4}{4}$ and, evaluated between 0 and z/3, gives $\frac{z^4}{324}$. Integrating $\int_0^3 \frac{z^4}{324} dz= \left.\frac{z^5}{1620}\right|_0^3= \frac{243}{1620}= \frac{3}{20}$. Multiplying by $2\pi$ from the first integral that is $\frac{3\pi}{10}$.

"Changing the order of integration" does NOT mean "changing form cylindrical to Cartesian coordinates". Here, since, again, we could do the $\theta$ integral separately, we are looking at $\int_0^3\int_0^{z/3} r^3 drdz$ That means that we are letting z go from 0 to 3 and, for each z, letting r go from 0 to $z^3$. We can visualize that as the region under the cone $r= z^3$ but inside the cylinder r= 27. We could also cover that region by taking r from 0 to 27 and, for each r, z going from $\sqrt[3]{r}= r^{1/3}$ to 3:
$\int_0^{27} \int_{r^{1/3}}^3 r^3dzdr$.

before I put in a homework pdf I have

$\tiny{244.15.7.7}$
$\textsf{Changing the Order of Integration In Cylindrical Coordinates}$
\begin{align*}\displaystyle
dV&=\int_{0}^{2\pi}\int_{0}^{3}\int_{0}^{z/3}r^3 \, dr \, dz \, d\theta\\
\end{align*}
$\textit{First, because neither the integrand}$
$\textit{nor the limits of the other two integrals involve$\theta$}$
$\textit{we can do that separately:}$
$$\displaystyle\int_0^{2\pi} d\theta= 2\pi$$
$\textit{so that is simply}$
$$\displaystyle dV= 2\pi \int_0^3 \int_0^{z/3} r^3 dr \, dz$$
$\textit{Then we can proceed with:}$
\begin{align*}\displaystyle
dV&= 2\pi\int_0^3
\biggr[\frac{r^4}{4} \biggr]_0^{z/3} dr \, dz \\
&=\int_0^3 \frac{z^4}{324} dz= \biggr[\frac{z^5}{1620}\biggr]_0^3= \frac{243}{1620}= \frac{3}{20}\\
&=2\pi \biggr[\frac{3}{20} \biggr]
=\color{red}{\frac{3\pi}{10}}
\end{align*}

1. What is the significance of the numbers 17.7.07 in this statement?

The numbers 17.7.07 most likely refer to a specific date or time, and are not relevant to the overall meaning of the statement. They may be a placeholder or part of a larger context that is not provided.

2. What does "other orders" refer to in this statement?

"Other orders" likely refers to alternative methods or approaches that can be used in a scientific experiment or study. These may be different from the usual or expected methods, but can still be effective and valuable in evaluating certain aspects of the research.

3. Can you provide an example of when other orders may be easier to evaluate?

For example, if a scientist is studying the effects of a new drug on a specific population, they may use traditional quantitative methods to measure the drug's effectiveness. However, they may also choose to use qualitative methods, such as interviews or surveys, to gain a deeper understanding of the participants' experiences with the drug.

4. How can scientists determine which orders will work well for their research?

The choice of which orders to use will depend on the specific research question, the available resources, and the goals of the study. Scientists may consult with colleagues, conduct a literature review, or pilot test different orders to determine which will be most effective for their research.

5. Are there any potential drawbacks to using other orders in scientific research?

While other orders can provide valuable insights and data, they may also introduce biases or limitations. For example, qualitative methods rely on the researcher's interpretation and may not be generalizable to larger populations. It is important for scientists to carefully consider the potential limitations and biases before incorporating other orders into their research.

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