Let √(x) = sin(u)
Differentiate both sides
= 1/(2*√(x))dx = cos(u)du
dx = cos(u)du / 1/(2*√(x))
= cos(u)du / 1/[2*sin(u)]
Invert and multiply
∫√(x)dx / √(1-x)
= ∫sin(u ) * cos(u) * 2*sin(u) / √(1-sin(u)^2) du
= ∫ 2*sin(u)^2 * cos(u) / √(1-sin(u)^2) du
= ∫ 2*sin(u)^2 * cos(u) / √(cos(u)^2) du
= ∫2*sin(u)^2 * cos(u) / cos(u)du
= ∫2 * sin(u)^2 du
= ∫2*[1 – cos(2u)]/2du
= ∫ [1 – cos(2u)]du
= ∫(1)du – ∫cos(2u)du
= u – (1/2) * sin(2u) + C
u – (1/2)(2) * sin(u) * cos(u) + C
u – sin(u) * cos(u) + C
= sin-¹(√(x)) – √(x) * √(1-x)”
Green’s Theorem, Triangle, x^2y^2+4xy^3
Hello again everybody. Tom from everystepcalculus.com and everystepphysics.com.
Calculus problem regarding a tangent line with a curve. Dealing with two variables. I show you how that works on my programs. Index 8 to get to my menu. I’m already at equation of a tangent line here to save time. And you’ll scroll down to it and then you’ll enter the function Alpha, you have to press alpha before you enter anything into these entry lines. Alpha x cubed plus 6 times x times y plus y squared minus 8 equals zero. You always have to transpose everything on the other side of the equation here and bring it over the left side and make zero on the right or a number on the right either one would work. Now because you’ve entered y in here this is going to be an implicit differentiation problem. So here’s the formula here that you just entered. In the implicit system. And it looks pretty good. I say it’s ok. Differentiate both sides of the equation with respect to x. So here we go both sides. We’re going to do each one individually. This one here because xy, we need the product rule. So you can write the product rule down and do it like this or you can just keep clicking and you’ll finally get to the product rule answer to which is this right here and 6x dy dx and we keep going. y squared is equal to this because it’s a y dy dx also. And you combine the functions, dy dx here and then no dy dx just regular X Y and the implicit answer is this right here minus 3x squared minus 6x, etc. Division sign here, you know. Now we’re going to evaluate it at a point and the point is alpha 1 and alpha 1. So xy is 11. I say it’s okay. We enter those variables into the numerator and the denominator, come up with a slope of minus 98. Slope of the line and then we figure that y equals mx plus b which is here is the slope, m, and then you add the 1 for the x and 1 for the y and then figure out what b is here. Subtract this from 1, 1 of course 8 eighths and you subtract you get 17 eighths. Subtracting a minus so you get 17 eighths. Here’s the answer y equals mx plus b. Pretty neat, huh? everystepcalclus.com you can go, and I also give you that angle of the line is pointing, notice that it’s pointing in this direction not up but down to the right and if you want to do it and other points, it’ll take you back there. Pretty neat, huh? everystepcalclus.com Go to my site, buy my programs if you want to pass calculus and subscribe if you want to see more videos that I might make you. Have a good one!
A man 5 feet tall walks at a rate 5 feet per second away from a light that is 16 feet above the ground. When he is 8 feet from the base of the light, find the rate at which the tip of his shadow is moving.
Hello again everyone, this is Tom from everystepcalculus.com and everystepphysics.com. We’re gonna do a triple integral from Calculus 3 right now. this is an example of Patrick JMT, my favorite instructor on the Internet and youtube. So I’m gonna show you how it works on my program. I don’t know anybody can do that problem, he can do it because he’s a genius, but for us students, etcetera, how do we do it? So let’s get started. Index 8 to get to my menu. I’m gonna scroll up because I can go to the bottom of the menu then, instead of going down quicker to the T section and we’re gonna choose triple integral. And we’re gonna enter our function, you have to press alpha before you enter anything into these entry lines here in my programs, okay. Alpha X times sin of y. I always show you what you’ve entered you can change it if you want. And we’re gonna use the order of integration, which is dx, dz, dy, which is in the example. You have the other choices in case that’s given on a test also. region cue enter these. And we’re gonna enter the region, q. We’re gonna enter these limits. This is alpha 0 for the x one. Alpha square root of 4 minus Z squared. Made a mistake so I gotta go back. Choose number 2. Alpha 0. Alpha square root of 4 minus z So here’s what you write on your paper That’s better, say it’s okay. Next one for the y is alpha 0, alpha pi. That looks okay. and alpha 0 for the z. alpha 2. That’s okay. So here’s what you write on your paper, the way you write a triple integral with dx, dz, dy order of integration. Here’s the function in here. We’re gonna do the dx first, and you put this over here with these lines, when you’re doing the range over this integration here. And here’s the integral of the first here. Of the first function. And if x equals the upper range… I show quotation marks here but you put parenthesis in there because you’re substituting this amount for any X in the integral, and it equals this, minus sign etc. And then we do the lower integral. X equals zero, and there’s 0 plugged in, you put parentheses around this instead of quotation marks, okay. And here’s the answer, upper range minus the lower range equals this right here. So that becomes the new integration function and I show you that here. Dz, dy is left, okay? So now we integrate that, come up with this, minus sin z, z squared, et cetera and over this range here, 0, 2. Add z equals 2. Here’s the answer here, at z equals 0 plus these in for all the z’s in the problem. And the answer is this, the upper range minus the lower range is 8 sin y divided by 3. We’re gonna use that for the integration function with the range of 0 and pi. At y equals pi minus 8 cosine, here’s the 8/3. Y equals zero, you plug that in here, you get minus 8/3. Upper range minus the lower range, notice the minus times the minus, you can’t remember that stuff a lot of times. Turns out to be, the volume is 16/3. Okay. New problem. So go to my site, subscribe so you can see other videos I might make. Or you can go to the menu on my main site and go scroll down to what you need to learn. And see my program works, because it sure teaches you quicker than a book or anything else. Okay, so have a good one.
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Hello, everystepcalculus.com. A problem from a student regarding Shell Method and the vertical axis system. I’m gonna show you how my programs do that one. I’ll do the X axes in another video. Index 8 to get to my menu, scroll up because we got to the bottom on the menu, and get to the S section, which is Shell Method, easier, quicker. Here’s Shell Method, and get a choice of x axes or vertical axes, I’m gonna do the vertical axes now. Number 2, there’s a formula for it, and there’s always two functions given. If not, you have to enter the second, which is generally 0. You have to press alpha before you enter anything in these entry lines here. Alpha, 2, times X plus 3 is the function given. And the other one we’re going to put is alpha 0. I will show what you’ve entered, you can change it if you want. I see it’s okay. Are limits given? Yes, they are given. Lower limit is alpha 1, upper limit is alpha 2. I’ll also show you that in case you made a mistake. See, that’s okay, and you get the radius of x, and the height is 2x+3, p(x) and h(x) are the functions. So we substitute that into the formula, which is right here. You had it just like this on your paper. And we’re gonna multiply the two functions together, we get this right here with the limits of 2 over 1. And now we do the integration which equals this right here over those limits. And X=2, you substitute 2 for all the X’s in the function. I have to use quotations here because that’s where the calculator does it, but you’re going to use parentheses around these 2’s here, because you’re substituting 2 for every x. That equals this which equals 68 pi over 3. And X=1, you’re gonna substitute 1 for all the x’s in there, which equals 13 pi over 3. Upper minus lower is 55 pi over 3. That’s the answer! Have a good one!