1 minute integral vs. 9 minutes integral, trig sub, calculus 2 tutorial

Learn trig substitution with this calculus tutorial! We will go over the integral of sqrt(9-x^2) and the integral of sqrt(9-x^2) from -3 to 3.

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#blackpenredpen #math #calculus #apcalculus

[SciDiFuoco]

I looked at the second one geometrically and it turns out that the first part (the arcsin) is the area of a circular sector and the second one (basically 1/2xy) is the area of a triangle. The sum of these two areas gives the value of the integral

[BriTheMathGuy]

Area of semi-circle is a great technique for these. Calc 1 (before trig substitution) professors sometimes throw this on tests!

[DanielGonzalezL]

I like the fact that this video is 11 minutes, so it indeed is a 1 minute vs 10 minute integral 🤣

[mikety]

For the indefinite integral, it is a squarish-bottomed slice of a circle bounded by a diameter and then two lines perpendicular to it. Such a figure can be decomposed into a "circular cap" (area bound by a chord) and a trapezium.

[ffggddss]

Both can be done geometrically; i.e., without calculus. The definite integral is just a semicircle of radius 3, and so, is:
∫₋₋₃³ √(9–x²) dx = ½π·3² = 9π/2

The indefinite integral can be evaluated (starting from x=0 and adding a constant of integration at the end) as a right triangle + circular sector:
r = 3; y = √(r²–x²); θ = sin⁻⁻¹(x/r)
∫ √(9–x²) dx = ½xy + ½·r²θ + C = ½x√(9–x²) + (9/2)sin⁻⁻¹(x/3) + C

Aha! I see now that I've been scooped by SkiFire13. Kudos to him/her!
I promise I didn't see that before I answered; I've done this sort of integral before.

Fred

[bulldawg]

Again, outstanding presentation ... I wish I had instructors like you in my college years back in the Stone Age :)

[jroseme]

Thanks bro. That double angle identity coming back to haunt me a decade later!

[ricardomembreno]

I am from Nicaragua, your canal is very great

[snejpu]

Now we can proof that 9/2*arcsin(1) + 0 - 9/2*arcsin(-1) - 0 = 9/2*pi, and so arcsin(1) - arcsin(-1) = pi. How cool is that. : )

[woodwardscreditcard]

I like how my course book has this as it's second test problem on integral substitutions when you're just trying to understand the very basics on even how to get started.

[saharhaimyaccov]

It's cool ..
On another topic, I found a quadratic equation that can find the levies in the right triangle when raising height to the rest:
c²x²-c³x² + (ab) ².
  You can record ab = ch and save a torch in two disappearances.
The square beacon yields two results, the two levies that stand on the rest.

[cajintexas]

Whoever looked at an integral like that and realized you could solve it by substituting a trig function was a straight-up genius. I can't even imagine making that connection.

[spiderjerusalem]

Thanks dude! I've been learning integrals from you, because i think it's interesting. I'm still 14 years old, hopefully i can master all of these stuffs.

[anubhavjain]

We have a formula for root(a square-x square) but I never bothered to learn how it was derived. Thanks to you, now I know and won’t forget the formula :)

[aimadelouardachi]

I am from moroco, Your canal is a very nice .

[kaartiknayak]

Sir i think there is a formula for integration of rt of a^2-x^2 which is given by x/2 rt (a^2-x^2) +a^2/2 sin inverse of x/a

[GDLameGames]

Thank you for using the Doraemon theme song. Brought back memories :')

[Shivam-mqde]

The 10 min can also be solved in 1 min, using a formula which can be derived by geometry or algebraically.

[lucasdepetris]

Hi, It would be nice if you make a video explaining the issue of why does everyone prefers "positive stuff", like positive roots, or cancelling square roots with seconds powers, being the second power inside of the square root. All that things.

[wollyculiao]

Is nice to look that the integral is well defined in [-3, 3] and the argument in arcsin is ×/3 (domain of arsin is [-1, 1]) solving any problem with the value of x