Integrals Definition and 1000 Threads

Homework Statement a region that resists a body of water follows the curve y=.3x^2 from 0<=y<=240 using water density of 1000 kg/m^3 and g of 9.8 m/sec^2 Homework Equations 0 to 240∫g(rho)(240-y)2(sqrt(y/.3) The Attempt at a Solution 9.8(1000)(2)(240^(3/2)(292.199-.730297(240)) i...

Homework Statement ∫(2x+1)/(x²+2x+1)(x²+x+1) Homework Equations none The Attempt at a Solution I've foiled this out to look like: ∫(2x+1)/(x^4+3x³+4x²+3x+1) I'm trying long division here but it's getting really ugly really fast. Should I foil this out in the first place or...

For any integral where the integrand is of the form f(θ)^z, with z a complex number, and f(θ) = sin(θ), cos(θ), tan(θ), ... etc., θ being either real or complex. Is it possible to explicitly solve for an antiderivative? I'm not aware of any such way I could use residues/series representations...

Homework Statement integral of dx/((9-(x^2))^(3/2)) A = 0, B = 3/2 Homework Equations Trigonometry Substitutions 3. The Attempt at a Solution : I am stuck with this question. So far, I got (1/9)integral of (1/cos^2(θ)) dθ

Homework Statement Evaluate ∫ F ds over the curve C for: a) F = (x, -y) and r(t) = (cos t, sin t), 0 ≤ t ≤ 2∏ b) F = (yz, xz, xy) where the curve C consists of straight-line segments joining (1, 0, 0) to (0, 1, 0) to (0, 0, 1) Homework Equations The Attempt at a Solution a) I first found the...

hey all i know and understand the component of curl/line integral relation as: curlF\cdot u=\lim_{A(C)\to0}\frac{1}{A(C)} \oint_C F\cdot dr where we have vector field F, A(C) is the area of a closed boundary, u is an arbitrary unit vector, dr is an infinitely small piece of curve C my...

Homework Statement Find the mass and center of mass of the lamina that occupies the region D and has the given density function ρ. D is bounded by the parabolas y = x^2 and x = y^2; ρ(x, y) = 23√x Homework Equations m = \int\int_{D} ρ(x, y) dA x-bar = \int\int_{D} x*ρ(x, y) dA y-bar =...

Homework Statement Let n>1/2 and consider the function f(x)=x^{-n} for x\in[1,∞) Calculate the volume of the solid generated by rorating f(x) about the x-axis, showing all details of your working. Homework Equations Since it is rotated about the x-axis, its axis of symmetry is...

I can do this calculation using different methods; my interest is improving my skills at using this method, rather than the answer. Trying to find the area of the ellipse \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 From the Jacobian, we get dxdy = rdrd\theta So I go from the above equation of the...

I quote a unsolved question posted in MHF by user poorbutttryagin on February 5th, 2013.

Homework Statement I want to know if I've gone about setting up these integrals in these questions properly before I evaluate them. (i). Find the mass of the cylinder S: 0 ≤ z ≤ h, x^2 + y^2 ≤ a^2 if the density at the point (x,y,z) is δ = 5z^4 + 6(x^2 - y^2)^2. (ii). Evaluate the...

I was wondering: Is there an even more general formula for the integral than int(x^k) = (x^(k+1))/(k+1) that accounts for special cases like int(x^(-1)) = ln|x| and possibly u substitutions?

I am unsure whether I have properly performed the integration of the integral ∫((sin(√x))^3*dx)/√x When I used my TI-Nsprire CAS to take the derivative of my answer in order to check if I was correct, and it came out differently. Now I used some trig identities to manipulate the problem, so I...

Hi, I shall show the following: (f*g) \star (f*g) = (f\star f)*(g\star g) where * denotes convolution and \star cross-correlation. Writing this in terms of integral & regrouping: \int_{\phi} \left(\int_{\tau_1} f(t - \tau_1) g(\tau_1) d\tau_1\right) \cdot \left(\int_{\tau_2} f(\tau_2)...

Hi, this is rather short one. I'm wondering if the two in the image below are equivalent. http://gyazo.com/62449de717e91b412f42771fd7d0a1af To me they appear to be equivalent, and I can't recognize any exceptions or trivial cases. Also, if it is equivalent would be it be acceptable to...

Homework Statement I have function f which is defined upon an interval [a,b]. I have calculated the mean value using the theorem \frac{1}{b-a} \int_{a}^b f(x) dx What I would like to do is to plot in Maple the mean value rectangle. Where the hight of this rectangle represents the mean value...

So, like i said in the Title this more of a theorycal question. In my university notebook i have written that an integral to converge has to happen the next: 1. The f has to be bounded (if not its just a dot) 2.The interval has to be finit. [THIS IS WHAT IT'S WRITTEN IN MY NOTEBOOK] See, my...

In most calculus textbooks, they use double integrals to evaluate the Gaussian integral. Where did they get the idea - or how did they choose the two variable function e^{-(x^2+y^x)} to evaluate it? I guess this is related...but if you were given a fairly hairy integral and it was suggested...

Homework Statement Write out the triple integral for the volume of the solid shown in all six possible orders. Evaluate at least 2 of these integrals. Homework Equations I attached a picture of the figure. The front : x/2+z/5=1...

Homework Statement Determine how many integrals are required for disk, washer, and shell method.Homework Equations x=3y^2 - 2 and x=y^2 from (-2,0) to (1,1) about x-axis.The Attempt at a Solution Since there are no breaks or abnormalities in the graph it appears that 1 integral will solve for...

I am in Calculus 2 and we're just reviewing calc 1. Can someone break down the concept of differentiating definite integrals for me? I am mostly struggling on the trig functions. The problem I am stuck on is ∫^{}x_{}0 cos(t^{}2) dt

Homework Statement Show that: I = \int\int_{T}\frac{1}{(1 + x^{2})(1 + y^{2})}dxdy = \int^{1}_{0}\frac{arctan(x)}{(1 + x^{2})}dx = \frac{\pi^{2}}{32} where T is the triangle with successive vertices (0,0), (1,0), (1,1). *By transforming to polar coordinates (r,θ) show that:* I =...

It seems like we calculate integration by doing the reverse of derivation. Differentiation is basically just using short-cuts for differentiation by first principles (e.g. power rule). If integration by first principles is the Riemann sum, then why don't we use short-cuts of the Riemann sum to...

Homework Statement Find the volume which lies below the plane z = 2x + 3y and whose base in the x - y plane is bounded by the x- and y-axes and the line x + y = 1. Homework Equations I = \int\int_{R} f(x, y) dydx = \int^{b}_{a}\int^{y=y_{2}(x)}_{y=y_{1}(x)} f(x, y) dydx The...

Homework Statement Not sure if this goes here or physics, but this is for my calculus 2 class so I decided here would be best. #22 Homework Equations W=∫Fdx The Attempt at a Solution I think the limits of integration are from 0 to 1 since the water is right under the spout, but...

It's been a year since I took Calc I, and I'm taking Calc II online this semester. This is technically a review problem from Calc I, and I managed the other seven, but I can't figure out how to solve this problem. 1.a Homework Statement ∫(a*sin(14x))/(\sqrt{1-196x^2} dx, evaluated at x=0...

1. Gaussian Please help me calculate some Gaussian integrals in the attached file which are used in my QFT calculations. Thank you very much!

I am doing some calculations in QFT. And, in my calculations, I have to deal with 5 Gaussian integrals as followed. Please help me calculate those 5 integrals. Thank you very much!

Homework Statement Despite the fact that this started as an extended AP Physics C problem, I turned it into a calc problem because I (sort of) can. If it needs to be moved please do so. There is a hollow solid sphere with inner radius b, outer radius a, and mass M. A particle of mass m...

Can we use Newton's method to approximate the value of definite integrals? (Thinking) EDIT: Ignore if the question doesn't make sense (which it probably doesn't).

Initial improper integral: ∫ dx / (1+x**2) * (1+ atan(x)) , x = 0, ∞ Substitutions: μ = 1 + atan(x) dμ = dx / (1 + x**2) μ(∞) = 1 + pi/2 μ(0) = 1 Integral: ∫ dμ / μ , μ = 1, 1+ pi/2 Then solve. I'm getting the right answer, but I think I'm botching something due to a lack of...

Homework Statement Using ∫∫kdA = k(b-a)(d-c), where f is a constant function f(x,y) = k and R = [a,b]x[c,d], show that 0 ≤ ∫∫sin∏xcos∏ydA ≤ 1/32, where R = [0,1/4]x[1/4,1/2]. Homework Equations ∫∫kdA = k(b-a)(d-c) 0 ≤ ∫∫sin∏xcos∏ydA ≤ 1/32 The Attempt at a Solution I tried to...

Homework Statement Find the volume bounded by the following surfaces: z = 0 (plane) x = 0 (plane) y = 2x (plane) y = 14 (plane) z = 10x^2 + 4y^2 (paraboloid) Homework Equations The above.The Attempt at a Solution I think it has something to do with triple integrals? But...

I understand that the indefinite integral is like infinite definite integrals, but how come when we calculate the definite integral we simply substitute the two values into the indefinite integral and subtract? Why do we subtract? Why not add? Also, there aren't the same thing, right? What's...

Homework Statement Homework Equations I need help solving intergral… \int \frac{dx}{(a+x)^2} The Attempt at a Solution I found the integral for… \int \frac{dx}{(a^2+x^2)} = 1/a arctan x/a But I don’t know how to apply that to the original integral which is a little different...

The combination of special relativity and quantum mechanics in a single framework makes our understanding of such systems to be true only in 4D, Minkowski space...I have noticed that recent published work concerning 2D systems and I am not sure about this reduction of 4D to only 2D, does it mean...

[EDIT]: Found the mistake, see the next post. Homework Statement Evaluate $$\iint_{S}{\rm e}^{x+y}dx\, dy,S=\{(x,y):\left|x\right|+\left|y\right|\leq1\} $$ 2. The attempt at a solution ##\left|x\right|+\left|y\right|## is the rhombus with the center at the origin, symmetrical about both...

Hi all, Thanks for response :) I Dont really understand what is surface integrals ?? and its difference with Surface Area using double integrals. Can anyone help ? thanks a lot...

Why does the incomplete gamma function have an imaginary component, when the exponential integral does not? \Gamma(0,z,\infty)\equiv\int^\infty_z \frac{e^{-t}}t dt Ei(z)\equiv-\int^\infty_{-z} \frac{e^{-t}}t dt Looking at how these integrals are usually defined I would have expected them to...

Homework Statement Let ## f: [0, a]## ---> ## \mathbb{R}## be positive and increasing. Prove that the function G, such that: ##G(x):= \frac{1}{x} \int_0^x f(t)dt## ## x\in (0,a)## is increasing. The Attempt at a Solution I know that if the first derivative of a function is positive, that...

Homework Statement Determine if the following integrals are convergent or divergent. Explain why. \int^{1}_{0} \frac{1}{1-x^{4}} dx The Attempt at a Solution I've tried using Comparison Test, using f(x) = \frac{1}{1-x^{4}} and\; g(x) = \frac{1}{1-x}, 0 \leq f(x)\leq g(x) in ] 0,1 [ and I...

I'm trying to understand how to use Taylor series expansion as a method to solve complex integrals. I would appreciate someone looking over my thoughts on this. I don't know if they are right or wrong or how they could be improved. I suppose that my issue is that I don't feel confident in my...

Folks, Is there a online source I can use for evaluating Incomplete elliptical integrals ##F(\phi,k)## and ##E(\phi,k)## I do not have Mathlab or Mathematica and Wolfram alpha requires payed registration for extended computation time. Any information will be appreciated. Regards

Hi, Referring to Jackson's Electrodynamics 3ed, page 197, line 5. He assumes that the magnetization can be divided into volume part and surface part, thus generating eqn 5.100. This is fine. In a straightforward way, I wanted to do the same but for electrostatics, eqn 4.32:∅= (1/4πε) ∫dv...

So I am trying to understand how and why the limits of surface and volume integrals come about. I think I came up with a easy to understand argument but not a mathematically sound one. Frankly its a little dodgy. Can anyone provide feedback on this argument or provide a better and possibly more...

Hi Evalute the following integers

Is it possible to do Gaussian integrals with half integrals. we would define then nth derivative of e^{-x^2} and then somehow use that. And this integral is over all space. any input will be much appreciated.

given ∫(2-5) f(x)dx=5 and ∫(4-5) f(x)dx=∏ , find a) ∫(5-5) f(x)dx = b) ∫(5-4) f(x)dx = c) ∫(2-4 f(x)dx = Im going over old tests of mine to get ready for my final, and I can't find anywhere in my notes how I solved this, I originally got (a. 0 b. ∏ c. 5-∏). Can...

Hi, Rather simple question here, just want to confirm: When we are dealing with indefinite double integrals, it's true to say ∫∫ f(x,y) dx dy = ∫∫ f(x,y) dy dx i.e, order of integration doesn't matter right?

Homework Statement Before the AP exam Cal Q Luss has 3 hours to cram: during this time, he wants to memorize a set of 60 derivative/integral formulas. According to psychologists, the rate at which a person can memorize a set of facts is proportional to the number of facts remaining to be...