Lorena Beard

2022-07-07

Find the antiderivative $x=\left(2/y\right)$
What is the antiderivative of $x=\frac{2}{y}$?
I have to find the antiderivative to do a volume problem, revolving around the y-axis.
I tried doing it, and I think it's $\frac{4{y}^{3}}{3}\pi$.

Alisa Jacobs

Expert

Step 1
Given the rule that $\frac{d}{dx}\left(\mathrm{ln}x\right)=\frac{1}{x}⇒\int \frac{1}{x}\phantom{\rule{thickmathspace}{0ex}}\mathrm{d}\mathrm{x}=\mathrm{ln}x+C$.
I would say that if you were integrating with respect to y, $\int \frac{2}{y}\phantom{\rule{thickmathspace}{0ex}}\mathrm{d}\mathrm{y}=2\mathrm{ln}y+C$.
If you wanted with respect to x, we need to rearrange the equation to make y the subject.
$x=\frac{2}{y}$
$y=\frac{2}{x}$
Step 2
Thus, similar to the 1st case,
$\int \frac{2}{x}\phantom{\rule{thickmathspace}{0ex}}\mathrm{d}\mathrm{x}=2\mathrm{ln}x+C$
Side note: in all cases except power of negative one we would use the reverse power rule as follows:
$\frac{d}{dx}\left({x}^{n}\right)=n{x}^{n-1}⇒\int {x}^{n}\phantom{\rule{thickmathspace}{0ex}}\mathrm{d}\mathrm{x}=\frac{{x}^{n+1}}{n+1}+C$
For you to obtain $\frac{4{y}^{3}}{3}\left(+C\right)$ you would be integrating $4{y}^{2}$ with respect to y. Thus I suspect some info is missing or the formula you gave might be wrong.

sweetymoeyz

Expert

Explanation:
Unless I'm mistaken, this depends on what variable it is being integrated with respect to.
Also, I'm not sure how you got pi in your answer. Once you've calculated the volume it would be appropriate, but just on the basis of the function $x=\left(2/y\right)$. I don't understand why integration would yield pi.

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