Integral of 1/cuberoot(x) from -1 to 1

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In this video I work out the Integral of 1/cuberoot(x) from -1 to 1. This problem is from an old book called Integral Calculus. It was written by H.B. Phillips and it was published in 1917.

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Thanks for showing us this, MS.

It's been a while since I took integral calculus. Getting from beginning to solution on this problem touched on a number of different issues with integration, so it was a particularly good review problem to bring back some memories.

greense

X^⅓ = -(-x)^⅓
The function is odd => symmetric integral about 0 equals 0.

alinaamosova

Nice video
Please upload proof videos...
I love proofs

hussainfawzer

A year ago you posted a video called "Pre Algebra Workbook #shorts" in which you say say you will provide a link to the book, but you did not give a link. So please post a link for the book or provide the author so I can look look up myself, thank you.

motomark

About that Philips solution and the dreaded skipped steps.

In the olden days, if a book presented a solution to an example like Philips did here, I would have lost my cool, tossed the book in the trash, and dropped out of the class (if I was taking one).

Now that I'm more mature--mathematically and just in life in general--I have a more nuanced take on step skipping by teachers and authors in their examples.

In the case of this Philips example, looking up the page, we see that the lesson the example was trying to present was about splitting an integral if there are undefined values in the interval the integral is being taken over.

It is fair to assume that Philips has already shown the reader how to handle the two resulting integrals--and probably done so recently--so the step skipping is okay. The reader should be able to say, "Oh, yeah, I know how to handle these."

And a reader with some mathematical maturity would then either (1) just buy in to Philips's solution and move on; (2) think a bit about the expressions being integrated and consider why it's reasonable that Philips got two offsetting amounts from his integration; or (3) actually do the integrals if still feeling a little shaky on the techniques required.

It is helpful for (2) and (3) that Philips did show the results of the integrations, rather than just jumping from the split to the solution of 0. Good on him for showing this step.
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A bit more about that nuance.

The more complicated the steps are, the further back they have been introduced (in time for lectures or pages & chapters for books--especially concepts from previous courses), and the reasonable expectation of mathematical maturity of the student/reader (algebra & precalc not much, calc some) are all factors that should influence the teacher or author's decisions about what solution steps to skip (or how much handwaving to do in a proof).

(Once the students have passed introductory courses in linear algebra and differential equations, hold onto your hats! A lot of step-skipping ahead.)

In short, it's about knowing your audience.
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Since MS was presenting this example with no preceding lesson--but only the reasonable assumption that this video was for an audience that had at some time done some integration--I was glad he showed all the steps he did. As I mentioned in another comment on this video, it was a nice review for that audience of a variety of issues that can show up when doing integrals.

And Philips seems to have known his audience--that there might be many who would do my (2) or (3) above--and showed those results of each of the split integrals to help them.

greense

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