Integrals Definition and 1000 Threads

Homework Statement Find the area of the part of z^2=xy that lies inside the hemisphere x^2+y^2+z^2=1, z>0 Homework Equations da= double integral sqrt(1+(dz/dx)^2+(dz/dy)^2))dxdy The Attempt at a Solution (dz/dx)^2=y/2x (dz/dy)^2=x/2y => double integral (x+y)(sqrt(2xy)^-1/5) dxdy Now I'm...

I think this be Analysis, I Need some kind of convergence theorem for integrals taken over sequences of sets, know one? Example, a double integral taken over sets such that x^(2n)+y^(2n)<=1 with some integrand. I'd be interested in when the limit of the integral over the sequence of sets is...

Use an appropriate volume integral to find an expression for the volume enclosed between a sphere of radius 1 centered on the origin and a circular cone of half-angle alpha with its vertex at the origin. Show that in the limits where alpha = 0 and alpha = pi that your expression gives the...

I started learning double integrals in this semester but I got some problems on it. Is it matter if the order of integration interchange? For example, V=∫f(x,y)dydx, can we rewrite the equation as V=∫f(x,y)dxdy? If not, could you explain on it little bit:sorry: Thanks!

Homework Statement I am after finding the centroid of the remaining area (hatched) when a circle is cut by a line. I made a diagram in CAD that demonstrates the problem. The idea is that, starting from the bottom of the circle, a cut is taken leaving a remaining shape whose area and...

I am reading Mark Newman's Computational Physics textbook. He goes over calculating integrals with Simpsons's Rule and the Trapezoidal Rule, and then he goes over calculating their errors. Why would I have to ever worry about the error of the integral? He has the chapters online at his website...

Homework Statement Evaluate the integral to find the area. Homework Equations The Attempt at a Solution[/B] gifs upload So I know how to find an anti-derivative for the most part. Here it's tricky because my equation has an exponent AKA square root. I tried to use the chain rule with...

Why isn't (5.298) the following instead? ##K(x, t_1; x', t_0) = \delta(x-x')\,e^{-\frac{i}{\hbar}H(t_1-t_0)}## My reasoning: Since \Psi(x, t_1) = e^{-\frac{i}{\hbar}H(t_1-t_0)}\,\Psi(x, t_0)\\ = e^{-\frac{i}{\hbar}H(t_1-t_0)}\,\int\delta(x-x')\Psi(x', t_0)\,dx' The exponential operator with...

Hello, I feel like I am fudging these integrals a bit and would like some concrete guidance about what's going on. 1. Homework Statement Evaluate ##I = \int_{-1}^{1} dx \delta'(x)e^3{x} ## Homework EquationsThe Attempt at a Solution [/B] I use integration by parts as follows, ##u =...

Homework Statement How do I find the surface area of a sphere (r=15) with integrals. Homework Equations Surface area for cylinder and sphere A=4*pi*r2. The Attempt at a Solution I draw the graph for y=f(x)=√(152-x2). A circle for for positive y values which I rotate. I will create infinite...

Homework Statement Find the upper, lower and midpoint sums for $$\displaystyle\int_{-3}^{3} (12-x^{2})dx$$ $$\rho = \Big\{-3,-1,3\Big\}$$ The Attempt at a Solution For the upper: (12-(-1)^2)(-1-(-3)) + (12-(-1))(3-(-1)) =74 For the lower: (12-(-3)^2)(-1-(-3))+(12-3)(3-(-1)) =42 For midpoint...

Homework Statement Change the Cartesian integral into an equivalent polar integral and then evaluate. Homework Equations x=rcosθ y=rsinθ I have: ∫∫r2cosθ dr dθ The bounds for theta would be from π/4 to π/2, but what would the bounds for r be? I only need help figuring out the bounds, not...

Hello guys, new member here. I've got a calculus project due Tuesday that I could use some help on. I won't bore you with the all details of the project, but first let's imagine an olive in the shape of a perfect sphere (with a radius always bigger than 6mm) that goes through a set of blades...

If I have a periodic wave x(t) with half-wave symmetry, it means that: x(t + T0/2) = -x(t) where T0 is the period of the wave. Would this automatically lead to the conclusion that X(t + T0/2) = -X(t) where X'(t) = x(t), i.e X(t) is the integral of x(t). ?

Hello, Since it was mentioned in my textbook, I've been trying to find Riemann's proof of the existence of definite integrals (that is, the proof of the theorem stating that all continuous functions are integrable). If anyone knows where to find it or could point me in the right direction, I...

Hey guys, need your help and hope someone takes the time: I need to solve the double integral by changing the order of Integration. I would really appreciate if you could illustrate the way of how to compute the solution. Best Estelle :)

Homework Statement Let ##f## be piecewise continuous from ##[0,+\infty[## into ##V = \mathbb{R} ## or ##\mathbb{C}##, such that ## f(x) \longrightarrow_{ x\rightarrow +\infty} \ell ##. Show that ## \frac{1}{x}\ \int_0^x f(t) \ dt \longrightarrow_{ x\rightarrow +\infty} \ell## Homework...

Hey folks, found a couple of interesting integrals and was able to solve one of them ANALYTICALLY! That means no numerical solutions needed. $$\int_{0}^{\frac{\pi}{2}} \frac{1}{1+(tan(x))^{\sqrt{2}}} dx$$ $$\int \frac{1}{1+e^{\frac{1}{x}}} dx$$ The first one I solved and will reveal analytic...

Homework Statement Let f(x) = x^2 and let P = { -5/2, -2, -3/2, -1, -1/2, 0, 1/2 } Then the problem asks me to compute Lf (P) and Uf (P). Lf (P) = Uf (P) = The Attempt at a Solution Please explain how to solve. I thought that L[f] meant to calculate the lower bound with respect to f(x)...

I've been teaching myself a little bit of Complex Variables this semester, and I had a question concerning complex integrals. If I understand correctly, then if a function f has an antiderivative F , then the line integral \int_C f(z) dz is path independent and always evaluates to F(z_1)...

Homework Statement Compute the volume of the solid bounded by the four surfaces x+z=1,x+z=−1,z=1−y2,z=y2−1 Homework Equations Fubini's theorem? The Attempt at a Solution I have tried to visualize this solid and define the limits; when I attempted to integrate by dxdzdy (in that order), I set...

Homework Statement Evaluating the following formula: The Attempt at a Solution Since the integral part is unknown, dividing the case into two: converging and diverging If converging: the overall value will always be 0 If diverging: ...?

Homework Statement \int_{0}^{2\pi} \dfrac{d\theta}{3+tan^2\theta} Homework Equations \oint_C f(z) = 2\pi i \cdot R R(z_{0}) = \lim_{z\to z_{0}}(z-z_{0})f(z) The Attempt at a Solution I did a similar example that had the form \int_{0}^{2\pi} \dfrac{d\theta}{5+4cos\theta} where I would change...

Homework Statement Two railroad tracks intersect at right angles at station O. At 10AM the train A, moving west with constant speed of 50 km/h, leaves the station O. One hour later train B, moving south with the constant speed of 60 km/h, passes through the station O. Find minimum distance...

If I want to integrate the volume inside a cylinder ##x^2+y^2 = 4R^2##, and between the plane (I think it's a plane) ##z= \frac{x^2+3y^2}{R}## and the xy plane, then I know how to convert it to cylindrical co-ords, find the limits of integration, and integrate r dr dθ dz. But exactly what am I...

Homework Statement Homework Equations Doesn't the integral of sec x tan x equal sec x? The Attempt at a Solution

Homework Statement Given: y = sqrt(x), y=0, x=3 Find the volume of the solid bounded by these functions, revolved around B) y-axis and C) line x=3. (Disk method) Homework Equations Disk method of finding volume using π ∫ r2dy The Attempt at a Solution Ok so, the part of the problem that I...

Hi! From this text: http://arxiv.org/pdf/nucl-th/0004061v1.pdf I need to resolve these integrals. 1) Equation (5), \int e^{-\omega_1/T}d{\vec k}_1 =? where \omega_1=\sqrt{m^2+{\vec k}^2_1} What function is K_2(m/T)? 2) Equation (26), \rho_s(T) Thanks!

Hello, When I recently was studying for my calculus I's rules of definite integrals, I was wondering if squaring a definite integral would be the same as integrating twice like in the following:( Definite integral of f(x) from a to b)^2 = definite integral ,from a to b, of the definite...

Homework Statement Use areas to evaluate the integral f(x)=5x+√(25-x2) on the following intervals a) [-5,0] b) [-5,5] Homework Equations ∫f(x) + g(x) = ∫f(x) + ∫g(x) also Area of a circle = pi(r)2 The Attempt at a Solution [/B] My first several attempts have centered around evaluating the...

Hi there! I was wondering if anybody knows what package or software CERN of other particle accelerators use to calculate their theoretical predictions. I need specifically tensor loop integrals of up to rank four and of three and four vertices. Thanks!

I'm reading through an undergrad physics book, and the author says he looked up the answer to the below integral in a table. I've tried to find tables of integrals with this integral included in them, but have failed so far. Can someone direct me to an exhaustive table of integrals and their...

I am currently taking calculus 3 and I am a little confused about the concept of double and triple integrals. Analytically, it's a breeze. I understand how to set limits, do all calculations, etc. What my question is, when I get an answer, what does the answer "mean"? For example, in this...

Hello, I was just introduced to this concept and I have solved a few problems, but I haven't come across any with denominators to a raised power yet. ∫ 1 / [(x+7)(x^2+4)] dx I would appreciate any directed help. 1. from the initial state I have broken the fraction into two assuming that...

bhobba submitted a new PF Insights post Some Useful Integrals In Quantum Field Theory Continue reading the Original PF Insights Post.

In the MVT for Integrals: ##f(c)(b-a)=\int_a^bf(x)dx##, why does ##f(x)## have to be continuous in ##[a,b]##.

Hi, Just curious: how does one define integrals over ##\mathbb R^{\mathbb R}, \mathbb R ^{\mathbb N} ##? I assume this must be a topic in Functional Analysis. I know a bit about abstract Wiener spaces; is there something else? And I assume the objects that are integrated are operators (linear...

Hi Physics Forums. I am wondering if I can be so lucky that any of you would know, if these two functions -- defined by the bellow integrals -- have a "name"/are well known. I have sporadically sought through the entire Abramowitz and Stegun without any luck. f(x,a) = \int_0^\infty\frac{t\cdot...

I have a few questions about the following property of definite integrals: $$\int_a^b f(x) dx = \int_a^c f(x) dx + \int_c^b f(x) dx$$ What exactly are the prerequisites for this property? Should ##c## be a member of ##[a,b]##? Should the function ##f## be defined at ##c##?

Homework Statement Hi, I am just trying to wrap my head around using path integrals and there are a few things that are confusing me. Specifically, I have seen examples in which you can use it to calculate the ground state shift in energy levels of a harmonic oscillator but I don't see how you...

So if we regard something as only being well defined if we can construct it, does this somehow affect what we think about integrals? The way I understand it, there is absolutely nothing in mathematics that tells you how to actually do an integral. Fundamentally, all we can do is cleverly pull...

I just did this following exercise in my text If C is the line segment connecting the point (x_1,y_1) to (x_2,y_2), show that \int_C xdy - ydx = x_1y_2 - x_2y_1 I did, and I also noticed that if we put those points into a matrix with the first column (x_1,y_1) and the second column (x_2,y_2)...

What is the difference between \int_{-\infty}^{\infty} \frac{x}{1+x^2}dx and \lim_{R\rightarrow \infty}\int_{-R}^{R} \frac{x}{1+x^2}dx ? And why does the first expression diverge, whilst the second converges and is equal to zero?

Homework Statement If ##f## is an even function then $$\int_{-a}^{0} f = \int_{0}^{a} f$$ Homework EquationsThe Attempt at a Solution My attempt was trying to show the upper sum of both integrals were equal. Take a partition of [-a,0] call it ##P_{1}##, and ##P_{2}## for [0.a]. if we can...

I'm not getting the right answer... why?

Do some one know how to integrate the Hermite(2n+1,x)*Cos (bx) and e^(-x^2/2), x from 0 to infinity?

I only memorized these trigonometric differential identities : `sin(x) = cos(x) `cos(x) = -sin(x) because I can convert tan(x) to sin(x) / cos(x) and sec(x) to 1 / cos(x) .. etcAnd there is no need to memorize some integral identities such as : ∫ sin(x) dx = -cos(x) + C ∫...

So I was asked to compute loop contributions to the Higgs and compute the integrals in spherical coordinates, I gave a look to Halzen book but did not found anything. Why, when and how to make that change?

Are these correct? Thanks in advance! 1.) Set up the triple integral for ##f(x,y,z) = xy + 2xz## on the region ##0 ≤ x ≤4, 0 ≤ y ≤ 2## and ##0 ≤ x ≤ 3xy + 1##. ##\displaystyle \int_0^4 \int_0^2 \int_0^{3xy+1} 2y +2xz\ dz\ dy\ dx## \text{2.) Set up the triple integral in cylindrical...

Homework Statement Let C be the straight line from the point r =^i to the point r = 2j - k Find a parametric form for C. And calculate the line integrals ∫cV*dr and ∫c*v x dr where v = xi-yk. and is a vector field Homework EquationsThe Attempt at a Solution For parametric form (1-t)i + (2*t)j...