Hello again, everyone. This is Tom from everystepcalculus.com and everystepphysics.com. I’m going to do a Triple Integral for Calculus 3 right now. This is an example of a Patrick JMT, my favorite instructor on the internet, on YouTube. So I’m going to 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,etc. How do we do it? Let’s get started. Index 8 to get to my menu. I’m going to scroll up because I can go to the bottom of the menu then instead of going down quicker to go to the T’s section. And we’re going to choose Triple Integral. And we’re going to enter our function. You have to press Alpha before you enter anything into these entry lines here in 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 going to 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 test also. And we’re going to enter region q. Enter these limits. This is Alpha 0 for the x one Alpha square root of 4 minus z squared I made a mistake so I gotta go back. Choose number 2. Alpha 0, Alpha square root of 4 minus z squared. Close up the parentheses. That’s better. I say it’s okay. Next one for the y is Alpha 0. Alpha pi. That looks okay. and Alpha 0 for z. Alpha 2. That’s okay. So here’s what you write on your paper. The way you write it with triple integral with dx dz dy order of integration. Here’s the function in here. So you’re going to do the dx first and you put this over here with these lines. Showing you’r doing a range over this integration here. And here’s the integral of the first function okay. And if x equals the upper range. I show quotation marks here but you put you put parentheses in there. Because you’re substituting this amount for an X in the integral. And it equals this, minus sin, etc. And then we do the lower integral. X equals 0 and there’s 0 and you put parentheses around this instead of quotation marks, okay? And here’s the answer, you have the 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, etc. over this range here 0 2. Add z equals 2 Here’s the answer here. And z equals 0. Plug these in for all the Z’s in the problem. And the answer is this. 8, the upper range minus the lower range is 8 sin y divided by 3. Now we’re going to use that for the integration function. With the range of 0 and pi. At y equals pi minus 8 cosine is 8 thirds.
Hello, everyone. Tom from everystepcalculus.com and everystepphysics.com. This is an everystepcalculus.com problem. And with my programs. Question on vectors. Let’s get started. Index 8 to get to my menu. Choose A and B vectors. This is a Calculus 3 problem. This student wanted to know how to find the magnitude of A times B. He could get the cross product but he do the magnitude of it. So, let me know you how to do that. You have to enter Alpha before you enter anything into these entry lines, here. So, Alpha minus 2. Alpha one. Alpha one. For the B Vector. Alpha 2, Alpha 1. Alpha 1. So we check see if that’s the vectors that you want. I say it’s okay. A times B is the cross product so we’re going to go to here, A times B. Matrix Multiplication. Write this on your paper, exactly as you see it. There’s the Cartesian System. Notating a vector. Right here. And here’s the answer for the cross product. Zero, four, four. So, we have one vector now which is 0 for X, 4 for Y, and minus four for Z. So now we’re going to go down and we’re gonna find new AB vectors. As we need to put a new vector in for A. So the answer was Alpha zero, Alpha four. That’s for the cross product. Alpha minus four. And we can just quickly go through these because there is no vector for B of A. So now we’re going to go up here and find magnitude of, magnitude of A, number 2. Choose that. This is the way you notate it, it’s got these two lines on the side. It’s called the magnitude of A. Do these calculations. Notice the square root of x square, y squared, z squared and we come up with 5.66 units. Pretty neat, huh? everystepcalculus.com. Have a good one.
Programs Included in Calculus 2 & 3 App
Hello, everyone. This is Tom from everystepcalculus.com and everystepphysics.com. This video is on a menu of Calculus 2 and 3 if you purchased that. What you get. I’m going to go over each item and show you. Index 8 to get to the menu. The busy sign here means that it’s loading the programs. Sometimes it takes longer because of the size of the program but after it’s loaded once, it’s very quick. For instance like this F1 8 to clear, I’m going to press it now. Notice how quickly it comes up right away. I’m going to go up to the top here. And start name the things. Look at this part up here as I name them. A and B vectors, that comes in Calculus 3, generally. You’re doing all sorts of things with adding vectors and subtracting vectors and doing everything possible with them. So A and B vectors, Acceleration, Angle of Vector, Arc Length, that’s the f of x system and then the r t system, Area of a Parallelogram, Component of A Direction of B, Cosine of A and B, that’s the cosine of the A and B vectors. Cross Product, dealing with vectors. Curl computing it at a point, Curl conservative, whether it’s conservative or not, Curl whether it’s divergent, and then the systems of notation which is MNP and PQR for the curl. There’s a double integral, definite integral xy and there’s a definite integral xyz. One is area one is volume. Disk method, Divergence at a point, dot product, double integral, eliminate the parameter, equation of a tangent plane, an equation of a plane, Gradient, Green’s Theorem, Integration by Parts, Interval of Convergence, Line Integral f of xyz, linear approximation 2 variables, linear approximation 3 variables, linear equation, natural log of x the derivative of that, natural log of x the integral, natural log of x solving for x, and then log problems, those are to different bases. Mass and spring of a wire, money, p and q points, parametric equation, partial derivative, partial fractions, partial fraction decomposition, path of objects, polar to rectangle, conversion, position vectors, projection of a on b, projection of b on a, ratio test and series when you’re talking about Taylor and Maclaurin series, rectangular to polar, rectangular box, that’s a standard computation in calculus. Series Convergence, Shell Method, Simpson’s rule, Sine of x cubed, cosine of x squared times. Sketch the graph when you’re talking about p and q points, speed which is the same thing as volume. Sphere center midpoint, Sphere equation, sphere radius, Surface area of Revolution, Surface Integral xy, surface integral xz, surface integral yz. Tangent of log of x, the derivative of that. Natural log of x. Tangent of plane to surface. Taylor and Maclaurin series, Trig and half angled formulas, Trig derivatives and identities, trig integrals, triple integral xyz, triple integral zyx, U Substitution, Unit Tangent Vector, Unit Tangent PQ divided by the magnitude which is PQ. That’s those two lines there on each side of the PQ. Unit Vector Opposite, Vector length, vector between p and q, vector magnitude, Vector, unit, tangent, and Vectors. Which is taking 2 or 3 vectors and finding the angles of those vectors, etc. Quite a series of those. Vector Field Divergent, Velocity, Volume of a parallelpiped, the washer method, and that’s it. Have a good one.
Hello, everyone. Tom from everystepcalculus.com and everystepphysics.com. Cross Product with regards to A and B Vectors. It’s a Calculus 3 Vector problem. Let’s do it. Index 8 to get to my menu. We’re going to scroll to Cross Product. Add the vectors. Alpha 8. You have to add Alpha before you enter anything into my menus. Alpha 9, Alpha minus 4. That’s vector A. Vector B is Alpha minus 7, Alpha 8, Alpha minus 3. I always show you what you’ve entered. You can change it if you want. I say it’s okay. We’re going to scroll down to Cross Product. There’s Cross Product. A times B cross product and B times A cross product. Whichever one, they’re different. We’re going to do A times B. And it’s a matrix situation, that’s for sure. So you write these down. Make sure that you keep these in the rows that you have. Make sure you write exactly what you see, here. All these equations. This is I J, and K. That’s a different vector format. 5, 52j, 127k. and here’s the vector situation with the arrows here. 5, 52, 127. Pretty neat, huh? That’s a tough problem if you don’t have a programs, if you don’t know what you’re doing. Especially on a test or something. Go to my site, subscribe so you can see more videos. Have a good one.