Find the antiderivative x = ( 2 <mrow class="MJX-TeXAtom-ORD"> / </mrow>

Lorena Beard

Lorena Beard

Answered question

2022-07-07

Find the antiderivative x = ( 2 / y )
What is the antiderivative of x = 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 4 y 3 3 π.

Answer & Explanation

Alisa Jacobs

Alisa Jacobs

Beginner2022-07-08Added 13 answers

Step 1
Given the rule that d d x ( ln x ) = 1 x 1 x d x = ln x + C.
I would say that if you were integrating with respect to y, 2 y d y = 2 ln y + C.
If you wanted with respect to x, we need to rearrange the equation to make y the subject.
x = 2 y
y = 2 x
Step 2
Thus, similar to the 1st case,
2 x d x = 2 ln x + C
Side note: in all cases except power of negative one we would use the reverse power rule as follows:
d d x ( x n ) = n x n 1 x n d x = x n + 1 n + 1 + C
For you to obtain 4 y 3 3 ( + C ) 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

sweetymoeyz

Beginner2022-07-09Added 8 answers

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 = ( 2 / y ). I don't understand why integration would yield pi.

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