Integrals Definition and 1000 Threads

Homework Statement The problem is to prove the work-energy theorem: Work is change in kinetic energy.Homework Equations Line integral stuff, basic physics stuff. The Attempt at a Solution I'm given the normal definitions for acceleration, velocity and I'm given Newton's second law. I'm...

In this link: http://math.berkeley.edu/~scanez/courses/math104/fall11/homework/hw10-solns.pdf For qustion 32.6, I'm not sure if I'm understanding how there can be a "sequence" of upper and lower darboux integrals. So (for example), what is the difference between U_{10} and U_{11}? Does...

Why is it that when the force field is z^2 and you take the surface integral over a sphere of radius a using spherical coordinates, that yields the flux to be (4pi a^3 )/ 3 BUT in a calculus book, the force field is z instead of z^2 evaluated using polar coordinates and it yields the same...

Homework Statement Hey guys, I just wanted to know, if it would be an incorrect assumption to say that greens theorem is directly correlated to a line integral. The reason I am assuming that is because the formula for a line integral in my calc text is...

What is the geometrical significance of the definite integral of a vector function if any? e.g. if you integrate a vector function that gives the velocity of some particle between t1 and t2, the vector we get indicates the distance traveled in the i, j and k directions right? does the...

Integrals...look easy but I'm still confused :( Homework Statement evaluate the integral ∫(36/(2x+1)^3)dx Homework Equations dx^n/dx = nx^(n-1) The Attempt at a Solution ∫(36/(2x+1)^3)dx = 6ln[(2x+1)^3]/((2x + 1)^2) ( I know this is wrong, but why??) ∫(36/(2x+1)^3)dx =...

I'm having trouble visualizing the riemann-stieltjies integral... Our textbook states: We assume throughout this section that F is an increasing function on a closed interval [a,b]. To avoid trivialities we assume F(a)<F(b). All left-hand and right-hand limits exist...We use the notation...

Homework Statement Let D be the triangular region in the xy-plane with the vertices (1, 2), (3, 6), and (7, 4). Consider the transformation T : x = 3u − 2v, y = u + v. (a) Find the vertices of the triangle in the uv-plane whose image under the transformation T is the triangle D. (b)...

Homework Statement This question is not the assignment problem but I think that if the result I mention here is true, then my assignment problem will be solved. Let (X,\Sigma,\mu) be a measure space. Suppose that (h_n)_{n=1}^\infty is a sequence of non-negative-real-valued integrable...

I have a differential equation of the form and I want to solve it using calculus, as opposed to using a differential equation method. \frac{d^2v}{dt} = \alpha where v is a function of t i.e., v(t) and \alpha is some constant. How do I solve for v(t) if the time ranges from t_0 to t...

Homework Statement Consider the change of variables x = x(u, v) = uv and y = y(u, v) =u^3+v^3 Compute the area of the part of the x-y plane that is the transform of the unit square in the 2nd quadrant of the u-v plane, which has one corner at the origin. (Since the transformation is 1:1...

I had a question on a quiz that I missed... I am unsure how they got this answer. If someone could explain it would be great! Write the integral that gives the length of the curve. y=f(x)=\int_{0}^{4.5x} \sin{t} dt It was multiple-choice(multiple-guess;)). \text{Choice A }...

I feel overwhelmed with something that should be capable of being explained very simply I think. Let's say you're getting thrown random questions involving surfaces/shapes creating boundries in ℝ3. Whats your step by step process in finding whether you want to do a double integral versus...

Homework Statement Compute ∫f ds for f(x,y)= √(1+9xy), y=x^3 for 0≤x≤1 Homework Equations ∫f ds= ∫f(c(t))||c'(t)|| ||c'(t)|| is the magnitude of ∇c'(t) The Attempt at a Solution So, with this equation y=x^3 ... I got the that c(t)= <t,t^3> c'(t)=<1,3t^2> I know that from the equation...

1. Ok, so the question is.. Find the exact volume of the solid bounded above by the surface z=e^{-x^2-y^2}, below by the xy-plane, and on the side by x^2+y^2=1. 2. Alright. So, I know that I can use a double integral to find the volume, and switching to polar coordinates would be simpler...

Homework Statement Step 1) I put the following into polar coordinates √(16-x2-y2)=√16-r2 Where r≤4 step 2 I solved for y in the original problem which is in the link y≤√(4-x2) step 3. I graphed the above function step 4. I put the above function in polar coordinates...

Hi, I was wondering about one particular example of this interchange. In Mallat's book, at the proof of Poisson Formula it's visible that the equation at the beginning of the 42nd page features the limit outside of the integral. It is my understanding that this limit had to be in...

Homework Statement evaluate the given trigonometric integral ∫1/(cos(θ)+2sin(θ)+3) dθ where the lower limit is 0 and the upper limit is 2π Homework Equations z = e^(iθ) cosθ = (z+(z)^-1)/2 sinθ = (z-(z)^-1)/2i dθ = dz/iz The Attempt at a Solution after I substitute and...

Homework Statement ∫∫e-(x^2+y^2) dA R Where R is the region enclosed by the circle x2+y2=1 First thing I did was graph the region where the function was enclosed. I saw that they didnt give a limitation to where the circle lied. So I automatically knew that d(theta) would lie on the...

I'm curious about the general solution to \int_{-\infty}^{+\infty} \exp[P(x)] dx Where P(x) is a polynomial in x with real coefficients and whose leading (highest) order is even and its leading order coefficient is negative. Intuitively these integrals ought to converge, but I'm having...

Homework Statement I need to solve this integral for a>0: \int _0^{\infty }\frac{\text{Sin}[x]}{x}\frac{1}{x^2+a^2}dx The Attempt at a Solution Using wolfram mathematica, I get that this integral is: \frac{\pi -e^{-x} \pi }{2x^2}=\frac{\pi (1-\text{Cosh}[a]+\text{Sinh}[a])}{2...

Hi all, Sorry if this is in the wrong place. I'm trying to understand probability theory a bit more rigorously and so am coming up against things like lebesgue integration and measure theory etc and have a couple of points I haven't quite got my head around. So starting from the basics...

So I was wondering if I defined a vector field F, and a Trajectory of a particle x=t y=.5at^2+vit+si and I can find the work done by the field on a particle moving on a path with a line integral ∫F.dr, so what would this equate to for a projectile does it apply to this?, could you give me a real...

Now I'm trying to get my head around this question. I just know they're going to give us a large degree question like this in the exam... Let's say: I = ∫[e^(x^2)]dx with nodes being x=0 to x=0.5 The 5th degree polynomial is 1 + x^2 + (1/2)(x^4) So my queries are: How would I go about...

Homework Statement Evaluate ∫f(z)dz around the unit circle where f(z) is given by the following: a) \frac{e^{z}}{z^{3}} b) \frac{1}{z^{2}sinz} c) tanh(z) d) \frac{1}{cos2z} e) e^{\frac{1}{z}} Homework Equations This is the chapter on Laurent Series, so I'm pretty sure...

Hi guys, I encountered it many times while reading some paper and textbook, most of them just quote the final result or some results from elsewhere to calculate the one in that context. So I'm not having a general idea how to do this, especially this one \int_k^\inf...

Hi all, I'm having trouble with finding an improper integral. The problem is ∫10(xln(x))dx My book says the answer is -1/4, but I do not understand how this is the case. lim(xlnx) as x approaches 1- = 0 lim(xlnx) as x approaches 0+ = ∞ So how does this value converge at -1/4? Thanks in...

How can I use complex numbers to evaluate an integral? For instance I'm reading a book on complex numbers and it says that to evaluate the integral from 0 to pi { e^2x cos 4x dx }, I must take the real part of the integral from 0 to pi { e^((2 + 4i)x) dx}. It totally skips how you do that. I...

$(1)\displaystyle \;\; \int\frac{1}{1+x^6}dx$ $(2)\displaystyle \;\; \int\frac{1}{1+x^8}dx$

Homework Statement Evaluate \int_{R} \int \frac{xy^2}{(4x^2 + y^2)^2} dA where R is the finite region enclosed by y = x^2\,\,\text{and}\,\, y = 2x The Attempt at a Solution I think the easiest way to integrate is to first do it wrt x and then wrt y, i.e \int_{0}^{4}...

I have a double integral: ∫∫sin^2(∏x/A)*sin^2(∏y/B)dxdy A=length along x B=length along y ranges: 0 to A(for x) & 0 to B (for y) Analytical result is: A*B/4 (unit^2) Now, I want to evaulate it numerically using trapezoidal rule. Infact, I have done it but not sure whether it is a right...

Homework Statement A 15 kg brick moves along an x axis. Its acceleration as a function of its position is shown in Fig. 7-32. What is the net work performed on the brick by the force causing the acceleration as the brick moves from x = 0 to x = 8.0 m? Homework Equations W = FΔx = max...

Homework Statement Find the amount of work (ω) done by moving a point from (2;0) to (1;3) along the curve y=4-(x^2), in the effect of force F=(x-y;x). Homework Equations The Attempt at a Solution ω = ∫((x-y)dx + xdy) ω = ∫(x-4+x^2)dx + ∫√(4-y) dy In the end, I get this...

I came across this question on a past paper and would appreciate some help. It is too hard for me at the minute. The problem - The volume of a body whose surface is formed on the underside by the paraboloid z = x2 + y2 and bounded on top by the cone z = 3-2(x2 +y2) (a) Explain which...

Hi, I have a question. In my calculus book, I always see the fundamental theorem for line integrals used for line integrals of vector fields, where f=M(x,y)i + N(x,y)j is a vector field.The fundamental theorem tells me that if a vector field f is a gradient field for some function F, then f is...

Homework Statement Find the derivative of: 1. f(x)=arccos(5x^3) 2. f(x)=∫cos(5x)sin(5t)dt when the integral is from 0 to x Homework Equations Chain rule, dy/dx=dy/du*du/dx The Attempt at a Solution For the first one, I can just take 5x^3 as u and then apply the chain rule...

Homework Statement I hope this is in the right forum, because this is a question on theory and not related to a specific problem. I was reading onlne about the Fundamental Theorem of Calculus. On one site the author wrote: F(x) = \int_{0}^{x} f(t) dt Later, he wrote: \int_{a}^{b}...

Let's say you have a function that is continuous in (a,b] but discontinuous at x=a and you integrate it from a to b. For example, \int^{1}_{0} \frac{1}{\sqrt{x}}dx I understand that the integral exists, and it can be easily computed by using the limit as x approaches 0 from the positive...

$f(-\theta) = |\sin(-\theta)| = |-\sin\theta| = |\sin\theta| = f(\theta)$. Hence, $f$ is even, and we need to only consider. $$ f(\theta) = \frac{a_0}{2} + \sum_{n = 1}^{\infty}a_n\cos n\theta. $$ $$ a_0 = \frac{1}{\pi}\int_{-\pi}^{\pi}|\sin\theta|d\theta = \frac{1}{\pi}\int_{0}^{\pi}\sin\theta...

While reading my calc book, I have developed a few questions about the situations in which definite integrals can exist. I've thought about these questions, and I feel that if I am able to answer some of them, I can make some other problems much easier, such as testing for convergence of a...

Homework Statement Given that n is a positive integer, prove ∫sin^n(x)dx=∫cos^n(x)dx from 0 -> pi/2 Homework Equations Perhaps sin^2(x)+cos^2(x)=1? Not sure. The Attempt at a Solution I honestly don't even know where to start. Should I set u=sin(x) or cos(x)? Doesn't seem to get...

I know that for a system of particles, the x, y and z coordinates of the centre of mass are given by (\frac{1}{M} \sum_{i = 1}^{n} m_i x_i, \frac{1}{M} \sum_{i = 1}^{n} m_i y_i,\frac{1}{M} \sum_{i = 1}^{n} m_i z_i). For a solid body, we can treat this like a continuous distribution of matter...

Is there a resource that is just a walkthrough of various kinds of problems one might get and the ways to solve them? I'm not talking about the basics from the calc and difEQ series (u substitution, partial fraction decomposition, trig substitutions, trig power reduction, integration by parts...

Hello, I recently learned about integrals in my high school calculus class. Yes, I know you add 1 to the exponent and divide the coefficient by that number, and the integral of a velocity over time graph gives the total distance traveled, but I have a different question: What is a quick and easy...

I posted an actual problem in advanced physics but no answer so i will try to get an math part answer from it. Suppose I have to solve this integral: I=\int {\vec{dl} × \vec A } Where \vec A = -\frac {1}{x} \vec a_{z} So it has only a z component and I have to find the vector cross of the...

Homework Statement http://pokit.org/get/img/65e8ba92c1d00bf7fc8be2b178757ed8.jpg If a=5b, and I1 and I2 are known, find the force on the triangular loop. Homework Equations The Attempt at a Solution For start, the field from the infinitely long wire is : \vec B=\large -\frac{\mu _{0}...

Hey guys and gals, this isn't actually an assignment of any sort, so I didn't want to put it in the homework section. This is also my first post, though I have been lurking for quite a while, reading the copious amounts of information available here. :p Anyhow, could somebody please elaborate...

Homework Statement In double integrals, the change of variables is fairly easy to understand. With u = constant and v = constant, along line KL v = constant so dv = 0. Therefore the only contributing variable to ∂x and ∂y is ∂v. The Attempt at a Solution However, in tripple...

Homework Statement calculate the contour integral \oint_{C} (y^2+ix)dz where C consists of the parabolic path z(t)=t^{2}+it for 0≤t≤1 followed by the straight line segment from the point 1+i to the point 0 Homework Equations The Attempt at a Solution so the contour is in 2 parts...

Can anyone point me to how to interpret Dirac notation expressions as wave functions and integrals beyond the basics of     <α| = a*(q)     |β> = b(q)     <α|β> = ∫ a* b dq For example in the abstract Dirac notation the expression     |ɣ> (<α|β>) can be evaluated as     (|ɣ><α|) |β>  ...