Integration Definition and 1000 Threads

Hi All, This is a very elementary question but, from calculus : ∫sinx dx from 0 to pi = 2, but i thought of it, in terms of the attached driagram. And if we think of it, this way, the integration is the area of a half circle of radius one and is equal to Pi/2. Where did i go wrong ?

Homework Statement Find area bounded by parabola y^2=2px,p\in\mathbb R and normal to parabola that closes an angle \alpha=\frac{3\pi}{4} with the positive Ox axis. Homework Equations -Area -Integration -Analytic geometry The Attempt at a Solution For p>0 we can find the normal to parabola...

I am bothered on how the area of circular sector was derived using integration. I have a solution but I don't know how it will go and come up with the formula: A = 1/2 sr and A = 1/2 r2 θ Can someone show me the solution for...

Homework Statement Im looking over the notes in my lecture and the prof wrote, \int_{0}^{2} \pi(4x^2-x^4)dx=\frac{64\pi}{15} Im wondering what's the indefinite integral of this equation. Homework Equations using u substitution The Attempt at a Solution \int \pi(4x^2-x^4)dx= \pi \int...

I'am trying to prove \int e^{ix}cos(x) dx= \frac{1}{2}x-\frac{1}{4}ie^{2ix} Wolfram tells so http://integrals.wolfram.com/index.jsp?expr=e^%28i*x%29cos%28x%29&random=false But I am stuck in obtaining the first term: My step typically involved integration by parts: let u=e^{ix}cos(x) and...

I'm wondering if this could be used to calculate the value of a contour integral directly. If a function has an antiderivative on the entire complex plane, this implies the field is conservative so we should always get the same value no matter which path taken. Shouldn't this also mean integrals...

Hello all, Here's something I've been trying to wrap my head around: In general, it seems that integration is 'harder' than differentiation. At least analytically. Numerically it may be the other way round. For one thing, it's often easy to differentiate implicit functions. For example...

Homework Statement imgur link: http://i.imgur.com/zGhum8Q.png The answer given in the back of the text is imgur link: http://i.imgur.com/hAprWPV.png Homework Equations v(t) = \frac{1}{C_{eq}}\int i_s dt The Attempt at a Solution I get the last piece of the piece-wise v(t) to be...

Hey everyone, So I've been learning about electric fields and all that good stuff. So now the question arises, what is the electric field at a point P away from an object (say a rod or a ring) at a distance D away? So we know E=Kq/r2 or E=KQ/r2 . The difference between q and Q: q the charge for...

I cannot understand the intergration done here The part how 1/a came, what happened to the x and how did tan come into this

Homework Statement Find area bounded by functions y_1=\sqrt{4x-x^2} and y_2=x\sqrt{4x-x^2}. Homework Equations -Integration -Area The Attempt at a Solution From y_1=y_2\Rightarrow x=1. Intersection points of y_1 and [/itex]y_2[/itex] are A(0,0),B(1,\sqrt 3),C(4,0). Domain of y_1 and y_2 is...

I am using the monte carlo method to calculate the integral of cos^2(x). This is done using N random numbers. In my case I am doing it for 10^i random numbers when i ranges from 1-6. I want to calculate the error on the integral for each value of i used. I know for each value of N I want to...

I am planning to measure the wing area of a Spitfire aircraft. I am going to use double integral formula, but firstly I need to derive the continuous function for the region of interest. Also, How to do that? I searched the entire internet and only found out the wing span of that aircraft...

In N D I want to do an integral of the flux through an N-1 D surface. The usual vector calculus integration theorems say I can integrate around the perimeter of the surface. OK, but that perimeter is now N-2 D. In 4D it could be a cube or a 2-sphere. I can't use Gauss' theorem because the...

This doesn't seem to me like an accurate characterization of NSA. In a synthetic treatment of the reals, we posit some axioms about the reals, and we simply assume that there exists a systems that obeys those axioms. In a constructive description of the reals, we build them up using Dedekind...

Homework Statement ∫sin(2sinh(3x)) Homework EquationsThe Attempt at a Solution okay so i did a u substitution letting u=3x so we get 1/3∫sin(2sinh(u)) but i have no idea how to get rid of the sinh, i tried writing in exponential form or maybe i have to use some identity.. I am not sure where...

Homework Statement Find the arclength of the curve x = ⅓(y2+2)3/2 from y=0 to y=1. Homework Equations (Please forgive the crazy definite integral symbols - I'm taking a LaTex class tomorrow so hopefully I'll be able to communicate more clearly from then on..!) arclength of curve = ∫ab √ (1 +...

Based on Schwartz inequality, I am trying to figure out why there should/can be the "s" variable which is the lower bound of the integration in the RHS of the following inequality: ## \left \|\int_{-s}^{0} A(t+r)Z(t+r) dr \right \|^{2} \leq s\int_{-s}^{0}\left \| A(t+r)Z(t+r) \right \|^{2} dr...

My problem : I have a function that I want to integrate, in the limit that a parameter goes to zero. I have a function ##f[x,r]## I want to compute ##F[r] = \int dx f[x,r]## and then series expand as ##r \rightarrow 0## This is impossible algebraically for me, but may be possible if I can...

Homework Statement Integrate $$v = \sqrt{2g\frac{T-v \pi r^2t}{\pi R^2}}$$ where g,T,r,R are constants Homework Equations N/A The Attempt at a Solution I tried playing around with the variables, but I am not sure how to integrate this. Just give me a little bit of hint would do. Thanks!

Homework Statement A plastic circular washer is cut in half and has a charge Q spread uniformly over it. If the electrical potential at infinity is taken to be zero, what is the electric potential at the point P, the center of the old washer? The inner radius of the washer is a, the outer...

Hello, I am currently in my first year of college, and I already took calculus in high school. I was able to solve all the problems, but I feel like I didn't understand everything conceptually. When integrating dy/dx=x you get, ∫x dx=1/2x2. But what exactly happened to the dx, why did it...

hi guys, i have a question. i saw this picture, and i don't really understand how they derived with the formula. The aim is basically to find the formula for the surface area of a spherical cap. why do you differentiate the x=sqrt(rˆ2-yˆ2)? how does that help to find the surface? and then...

Homework Statement a) ∫δ( x² cos x) dx with x from -∞ to +∞ b) Integral over C, with C = unit circle ∫C δ(α-α0) a ds a =(1,1) Homework Equations a) δ(g(x)) =∑δ(x-xi)/|g'(xi)| b)? The Attempt at a Solution a) g(x) = x² cosx g'(x) = 2x cos x - x² sin x xi = 0 and (2n+1)π/2...

Hello, I was wondering if someone out there could help me. The question: The speed, v metres per second, of a particle is given as v=8sin2t+3cos2t where t is in seconds. i) Find the total distance traveled in the time interval 0≤t≤1 ii) Find the average speed over this time interval.

Ok, I have encountered this one a few hours ago: \int \frac{cos x}{\sqrt{a - b cos x}}dx I have tried all techniques I know, integration by part, half-angle formula, substitution, trigo approach, and so on, but it seems, I could not integrate this one. I am no noob in integration, but this one...

Hi everyone, I am working on a problem related to pharmacodynamics, and I'm stuck with an integral that doesn't seem to have an analytical solution. Thing is, I'm not a mathematician, so maybe there is a solution and I can't see it. I wonder if anyone can please have a look and tell me. Here's...

It's my first post so big thanks in advance :) 1. Homework Statement So the question states "By interpreting <pxΨ|pxΨ> in terms of an integral over x, express <Ekin> in terms of an integral involving |∂Ψ/∂x|. Confirm explicitly that your answer cannot be negative in value." ##The 'px's should...

Homework Statement The anti-derivative of ∫##\frac{sinx}{sin^2x+4cos^2x}## is ##\frac{1}{\sqrt{3}}tan^{-1}((\frac{1}{\sqrt{3}})g(x))+C## then ##g(x)## is equal to : a. ##secx## b. ##tanx## c. ##sinx## d. ##cosx## Homework Equations ##d(cosx)=-sinx dx## The Attempt at a Solution I tried...

Homework Statement I was given this question as a part of an assignment and lost a mark because of a step.Homework Equations the integral of cos^5(x) dx after some fiddling and substitution it gets to this (1 - u^2)^2 du In the solutions there is a step that says refine = (u^2 - 1)^2...

Hi, I have an example in my studies as follows: If v = 200sin(100pi*t + 0.2) then evaluate Integral of v^2 dt between limits of 0.005 and 0. I have integrated it and used the double compound angle formula sin^2 A=1-cos2A and come up with the following as per the solution to the example in the...

Hello people! I have ended up to this integral ##\int_{φ=0}^{2π} \int_{θ=0}^π \sin θ \ \cos θ~Y_{00}^*~Y_{00}~dθ \, dφ## while I was solving a problem. I know that in spherical coordinates when ##\vec r → -\vec r## : 1) The magnitude of ##\vec r## does not change : ##r' → r## 2) The angles...

Hi, This is an example in "Barron AP calculus" I learned from some past threads that "dx" in integration either means △x which is a infinite number or indicates the variable with respect to which you're integrating. In the equation above, it seems that dx is multiplied by (1-3x)^2. Isn't dx...

I was not sure where to post this here or in calculus, but seeing as the underlying basic principle of my question is regarding parity of functions I am posting it here, but feel free to move if needed. Basically I am getting ready for a (intro to) QM exam and I still struggle with some basic...

Not a homework problem, so not sure if this is in the right area. Thought it was too sophomore to post in the maths forums. So, when you integrate y within defined limits for x, you get the area between the graph and the x-axis within those limits for x. But a) Why is this always between the...

Homework Statement I have included the LaTex version of the problem. \int \frac{sin^2 x}{1+cos^2 x} dx Homework Equations Simplifying fraction Partial fractions The Attempt at a Solution I have uploaded my attempt at the solution.

Homework Statement http://postimage.org/][/PLAIN] free picture upload 2. The attempt at a solution I want to go width times delta height. To do this I must describe width in terms of height. Here they used the Pythagorean theorem which is weird to me because I don't see a nice triangle...

Can someone explain why the slope dy/dx = 0 at x = L/2? L/2 is the midpoint and there would be a deflection here so surely the slope of the deflection curve shouldn't be 0? I'm finding it hard to visualise this. EDIT: I think I understand the above. The slope of the deflection curve at x = L/2...

I want to integrate a function numerically from 0 to infinity where for small x ##f(x)\sim x^{-5/2} \exp(-a/x)## and for large x ##f(x) \sim \exp(-bx)##. How do I best treat the steep rise for small x?

I'm having some trouble with this problem (adapted from K&K 1.23). Homework Statement An elevator is programmed to start from rest and accelerate according to \begin{align*} a(t) &= \frac{a_m}{2}\left[1 - \cos\left(\frac{2\pi t}{T}\right)\right] &\mbox{for }0\leq t\leq T \\ a(t) &=...

Homework Statement The probability that a particular computer chip fails after a hours of operation is given by 0.00005∫e^(-0.00005t)dt on the interval [a, ∝] i. find the probability that the computer chip fails after 15,000 hours of operation ii. of the chips that are still operating after...

Homework Statement ∫ sec2 (7-3θ) dθ Homework Equations ∫ cos(kx+b)dx= ((sin(kx+b))/k)+C ∫ sin (kx+b)dx=-cos " (k and b are constants) The Attempt at a Solution The answer in the book is: ∫ -1/3(tan(7-3x))+C. I assume by x they meant θ. The book only gave the two formulas above for solving...

Hi all, I have the average value of a function between limits of 7.3826 and 0 which equals 0.4453. I have used the formula for average value function and attached the equation I need solving as I don't know how to use the Latex commands. P is what I am trying to work out. Unfortunately I have...

Homework Statement Let: ##I=\int _{-1} ^{1}{\frac{dx}{\sqrt{1+x}+\sqrt{1-x}+2}}## Show that ##I=\int_{0}^{\frac{ \pi}{8}}{\frac{2cos4t}{cos^{2}t}}## using ##x=sin4t##. Hence show that ##I=2\sqrt{2}-1- \pi## Homework EquationsThe Attempt at a Solution The substitution is ##x=sin4t## which...

Just for my own entertainment I integrated the Lorentz factor with respect to velocity, using basic trig sub, I got the equation arcsin(v/c)*(mc^2). What does this mean? Is it just useless and irrelevant in the physics world or does it have some sort of hidden meaning?

How did they go from ∂Ψ/dθ = m to Ψ = mθ + f1(r) ? I mean I understand how they got mθ but can someone explain the f1(r) term? Similarly, how did they go from -∂Ψ/dr = 0 to Ψ = c1 + g1(θ) ? I understand how they got c1 but why the additional term? I can't provide further working out as...

Hi Community, I have this question and I stuck on the last question: (b) How many subintervals do you need to estimate the area to within 0.1units$$^2$$ I thought it was 40 subintervals but I am thinking that my answer is divisions along the $x$ axis. The other questions I am confident I...

Find the area of the indicated region. We suggest you graph the curves to check whether one is above the other or whether they cross, and that you use technology to check your answer. Between y = x^2 − 5x + 2 and y = −x^2 + 5x − 6 for x in [0, 4]

hey guys, I am stuck on a question which is The curve in the picture shown has equation y = bx(x − 2). (a) Find b given that the shaded area is 4 units 2 (b) Find the x−coordinate of the point A if the line OA divides the shaded area into equal parts. Ive done part (a) which...

Homework Statement i am done with this question and get the answer 37 but it required us to change the order of integrationHomework EquationsThe Attempt at a Solution i have no idea... i tried to do something like [/B] or but none of them can obtain the answer 37