Partial antiderivative/integration?
So I have the equation z(sub xy) =5xy^2 and am trying to find two possible formulas for z(sub y) which is the derivative of z in regards to y. Which I figure is the integral of z(sub xy) in regards to x. So i integrated z(sub xy) in regard to x and came up with z(sub y) = 5/2*x^2*y^2 and read online that when doing partial antiderivatives instead of having + constant (+C) at the end you have +g(x) which is plus a function. So I sort of know what one of the formulas could be, I could throw on a +g(x) onto one and them make up a g(x) for a second possible formula? any help would be greatly appreciated
Area between curves. Mainly to do with finding antiderivatives.?
I seem to be running into the same issue on two problems. I'm sure it's just something simple to work out, but..... y=e^x, y=xe^x, x=0 I have one value given (x=0), so I just need the other one. e^x = xe^x x=1 so, [0,1] Now I need to determine which is bigger/smaller. We'll use x=(1/2) y=e^x y=e^(1/2) ----- Bigger y=xe^x y=(1/2)e^(1/2) ----- Smaller so, A= Anti(b-top)(a-bottom) [f(x)-g(x)]dx A= Anti(1-top)(0-bottom) [e^x-xe^x]dx Now, I don't know how to take the antiderivative of that. Substitution won't work. Basically, I have the same problem here... y=cos(pix), y=4x^2-1 I cannot find the intersection by equating them because there is no way to isolate x. I must graph them. I get a domain of [-1,1] and that y=cos(pix)> y=4x^2-1 so, A= Anti(b-top)(a-bottom) [f(x)-g(x)]dx A= Anti(1-top)(-1-bottom) [cos(pix)-4x^2+1]dx Again, I can't seem to find the antiderivative of this. I'm sure it's something simple, or some stupid mistake I made.
How do I find the antiderivative of xe^(-.5x)?
This is done with Integration by Parts. That states that ∫ u dv = (u * v) - ∫ d du Let u = x. Then du = dx Let dv = e^(-x/2). Then v = (e^(-x/2) / (-1/2) So u*v = x * e^(-x/2) * (-2)..............Note: (1 / (-1/2)) = -2 and ∫ v du = ∫ (e^(-x/2) / (-1/2) dx = (-2) * (-2) * e^(-x/2) = 4 * e^(-x/2) So the answer is: x * e^(-x/2) * (-2) - (4 * (e^(-x/2))) Since both expressions have a factor of (-2 * (e^(-x/2))), we can simplify this, getting: [(-2 * (e^(-x/2))) * (x + 2)] + C <----------- Answer .
What are some real-life applications of integration and differentiation?
Motion is. On Limits, Standard Zeno's Dichotomy Paradox: If you wish to move, from point A to point B, 1 meters apart, you need to move to 1/2 meter first, and for that, you need to move 1/4 meter first and so on. How will you ever get started? You will, because once you define that zero is the limit of this sequence of numbers, and add the numbers, the sequence of sums, [math]1 + 1/2 + 1/4 + 1/8 + ... [/math] converges to 2. Without limits, you will have no way to explaining to Zeno that the sum of an infinite number of terms can be a finite number. On Differentiation, The most common example: Velocity. Defined as rate of change in displacement. Let's say you take 12 seconds to move from point A to B to C, 12 meters apart. Where were you at the 6th second? Did I mention these points do not lie on the same plane? And the velocity vector changes direction midway. This is a very simple example. Rate of change shines through as a concept when piled over itself. Acceleration - Rate of change of velocity. Jerk - Rate of change of acceleration. Jounce - Rate of change of jerk. Then they run out of words and just say nth order derivative :) The point is, there exists a way, and a standard dependable way at that, to calculate this stuff. On Integration, To keep true to my theme of using Motion as an example, take Brownian motion of Gas Particles. Something like this, Without integration, we wouldn't be able to explain the movement. More importantly, we wouldn't be able to predict these kind of movement with good enough probability. See: Path integral Understanding motion helps us Build - Cars and Concordes and Pianos and Pacemakers and Prosthetics and Processors and many other things. It doesn't get more real life than this.
What is integration in mathematics?
Integration is the act of bringing together smaller components into a single system that functions as one. In an IT context, integration refers to the end result of a process that aims to stitch together different, often disparate, subsystems so that the data contained in each becomes part of a larger, more comprehensive system that, ideally, quickly and easily shares data when needed. This often requires that companies build a customized architecture or structure of applications to combine new or existing hardware or software and other communications.MathematicsIntegration (Mathematics), the computation of a definite integral, a fundamental concept of calculus, which allows, among many other uses, computing areas and averaging continuous functions.Indefinite integration, in calculus, the process of calculating indefinite integrals, also known as antiderivatives.Symbolic Integration, the computation, mostly on computers, of antiderivatives and definite integrals in term of formulasNumerical Integration, the numerical methods for computing, usually with computers, definite integrals and, more generally, solutions of differential equationsOrder of integration, the number of times a time series must be first differenced in order to make it stationaryIf you like my answer, follow me.RegardsParul
Why do we use integration?
Integration is one of the most beautiful parts of calculus in my opinion.When I was first taught the formula of the volume of sphere I was floored because I had no clue how someone could come up with V=4/3πr^3. In 9th grade I decided to delve deeper. I wanted to know how they got there. Lo and behold, it's because of integration. You take half a circle’s equation in the plane and rotate it around the x axis and find the volume using the disk method. That may not have been too clear, but look at this:See that? It's a curvy snake-like graph right? I thought you could only find areas of nice looking shapes like squares and circles. Integration can let you find the area under each of those humps in the graph to exactness. Today, I see these curves as nice looking, and squares and circles are boring. Integration is beautiful in its own right. The real world uses of integration are really infinite. It's everywhere around us. We need it when we discuss quantum mechanics, for example. See that phone you're holding? It's a miracle of quantum mechanics. Transitively, integration caused your cell phone.
What is the definition of integral calculus?
integral calculus | Definition of integral calculus in English by Oxford DictionariesA branch of mathematics concerned with the determination, properties, and application of integrals.integral | Definition of integral in English by Oxford DictionariesA function of which a given function is the derivative, i.e. which yields that function when differentiated, and which may express the area under the curve of a graph of the function.Integral - WikipediaIn mathematics, an integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data. Integration is one of the two main operations of calculus, with its inverse, differentiation, being the other. Given a function f of a realvariable x and an interval [a, b] of the real line, the definite integral[math]{\displaystyle \int _{a}^{b}\!f(x)\,dx}[/math]is defined informally as the signed area of the region in the xy-plane that is bounded by the graph of f, the x-axis and the vertical lines x = a and x = b. The area above the x-axis adds to the total and that below the x-axis subtracts from the total.