Homework Statement
Show that T(u,v) = (u2 - v2, 2uv)
maps to the triangle = {(u,v): 0 ≤ v ≤ u ≤ 4} to the domain D
bounded by x=0, y=0, and y2 = 1024 - 64x.
Use T to evaluate ∬D sqrt(x2+y2) dxdy
Homework Equations
The Attempt at a Solution
x=u2-v2
y=2uv
Jacobian= 4u2+4v2 dudv
I guess the...
The ratio of integrals
∫〖a(x) b(x)dx〗/ ∫c(x) b(x)dx
can be maximized by choosing b(x) equal to the delta function at the point where a(x)/c(x) is a maximum.
Can anyone provide the solution for choosing b(x) when b(x) cannot equal the delta function, b(x) is greater than zero with a...
Homework Statement
do my professor did this in class, and it doesn't make sense to me
∫cos^5(x) sin^4(x) dx
∫cos^4(x) sin^4(x) cos(x) dx
∫(1-sin^2(x))^2 sin^4(x) cos(x) dx
∫(sin^4(x) -2sin^6(x) +sin^8(x))cos(x) dx
so the above I get but when my professor integrated I became lost, this...
1. The problem statement, all variables and given/known data
Calculate the given iterated integrals ∫02 dy ∫0yy2 * exy dxMy attempt:
∫20dy[exy*y]y0
= ∫20 ey*y*y - ex*0*0
= ∫20ey2*y dx
= [ey^2]*y]20 = 2e4
Is this correct?
Homework Statement
Evaluate the intergrals:
a) integral of 3^(-4*z^2) dz from 0 to infinity
b) integral of dx/(sqrt(-ln(x))) from 0 to 1
c) integral of x^m * e^(-a*x^n) dx from 0 to infinity
Homework Equations
gamma(n) = integral of e^(-w) * w^(n-1) dw from 0 to infinity
The...
In "Probability Theory: The Logic of Science", the author E. T. Jaynes relates that Prof. Hansen at Stanford evaluated integrals by treating constants like pi in an integrand as a variable. Sounds fantastic! Does anyone know how this is done?
Homework Statement
Find the volume of the solid enclosed between the cylinder x2+y2=9 and planes z=1 and x+z=5Homework Equations
V=∫∫∫dz dy dzThe Attempt at a Solution
The problem I have here is setting the integration limits. I first tried using:
z from 1 to 5-x
y from √(9-x2) to -3
x from -3...
Improper Integrals -- Infinite Intervals
Homework Statement
Evaluate the integral.
(from e to infinity) ∫(25/x(lnx)^3)dx
Homework Equations
The Attempt at a Solution
I know that for evaluating improper integrals, you can take the limit as t approaches infinity of the given...
Homework Statement
Suppose that p and q are points in U, where U is an open, path-connected, simply connected subset of Rn and c1 and c2 are smooth curves in Rn with c1(0)=c2(0)=p, c1(1)=c2(1)=q. Let w be a 1-form on U. Prove that the line integral of w over c1 equals the line integral of w...
Someone told me Isaac Newton developed some infinitesimal triangle series to find the area of a random blob, but I think there might be some way to do it this way by drawing many lines from a central point to the edge, although that would make more of a pie slice, but is there some way to...
Greetings chaps,
This will probably be old hat to most of you, but I'm beginning to start Quantum mech. so that I can develop a deeper understanding of its application in Chemistry ( I'm a Chemistry undergrad -gauge my level from that if you will!)
i.) First of all, would I be right in...
Homework Statement
Let I=[a,b], f : I to R be continuous and suppose that f(x) >= 0 . If M = sup{f(x):x ε I} show that the sequence $$\left( \int_a^b (f(x))^n \, dx \right)^\frac{1}{n}$$
converges to M
The Attempt at a Solution
Where do I start? I'm thinking of having g_n(x)=...
Please help me to evaluate the following integrals:
1) $\int\frac{x^{4}+1}{x^{2}+1}dx$
I recognise the form of $x^{2}+1$ in the denominator corresponds to an inverse tangent derivative. But how would I deal with the numerator in this respect?
2) $\int\frac{1}{x^{2}+x-6}dx$
I believe this...
I see equations of the form,
y=\int_{-\infty }^{t}{F\left( x \right)}dx
a lot in my texts.
What exactly does it mean? From the looks of it, it just means there is effectively no lower bounds.
I looked up improper integrals, but I can't say I really understand what is going on.
So when...
Homework Statement
As part of an assignment on matter wave diffraction I'm to calculate the following integrals
I_1=\int_0^{\infty}G(\vec r_2,\vec r_1;\tau)e^{i\omega\tau}d\tau,\quad
I_2=\int_0^{\infty}G(\vec r_2,\vec r_1;\tau)e^{i\omega\tau}\frac{d\tau}{\tau}
Homework Equations
To do so...
Ok, so for Q6, I first said that
z = 3 - 3x - 1.5y
Using (∂z/∂x)^2 = 9, (∂z/∂y)^2 = 9/4
I then did a double integral of (x + y + (3 - 3x - 1.5y)) * sqrt(9 + 9/4 + 1) dA
Letting y and x be bounded below by 0 as stated, and x bounded above by 1 - 0.5y and y bounded above by 2, I went...
Been a long time I had my integral class so I forgot almost everything I knew... I need to integrate to see if the serie converge (limn→∞ an = 0). Thus, there is a theorem of the integral, if you evaluate the limit of the integral of a serie when it tends to the infinite minus when x=1 you can...
Hi,
In my fluids work I have come to integrals of the type:
\int_{0}^{\infty}\frac{e^{ikx}}{ak^{2}+bk+c}dk
I was thinking of evaluating this via residue calculus but I can't think of the right contour, any suggestions?
Mat
$\displaystyle \int_{0}^{1} \ln \ x \ dx $ is not a proper Riemann integral since $\ln \ x $ is not bounded on $[0,1]$. Yet $ \displaystyle \int_{0}^{1} \ln \ x \ dx = \lim_{n \to \infty} \frac{1}{n} \sum_{k=1}^{n} \ln \left(\frac{k}{n} \right)$. Is this because $\ln \ x$ is monotone on $(0,1]$?
Hey people,
I need to calculate inner product of two Harmonic oscillator eigenstates with different mass. Does anybody know where I could find a formula for
\int{ H_n(x) H_m(\alpha x) dx}
where H_n, H_m are Hermite polynomials?
Homework Statement
Use
1) \frac{1}{2\pi}\int\limits_{-\pi}^{\pi} \frac{r_0^2 - r^2}{r_0^2 - 2rr_0cos(\theta-t) + r^2} dt = 1
to compute the integral:
2) \int\limits_{-\pi}^{\pi} \left[1 - acos(x) \right]^{-1} dx
for 0<a<1
[/itex].
The Attempt at a Solution
I looked on Wolfram...
Homework Statement
Evaluate the integral \int\int_S \sqrt{1+x^{2}+y^{2}}dS where S:{ r(u,v) = (ucos(v),usin(v),v) | 0\leq u\leq 4,0\leq v\leq 4\pi }
2. The attempt at a solution
Here is my attempt, I am fairly sure I am right, but it is an online assignment and it keeps telling me I am...
Homework Statement
I understand everything except why the derivative of 2 + tan (x/2) is .5 + sec^2(x/2)
I don't understand the .5 part. I understand the sec part. I would think the derivative of 2 would be C, or just disappear.
Homework Statement
The Attempt at a Solution
My book is not explaining very well the steps at solving these problems. There is a step that I'm missing
step 1. find 1/(b-a) easy
step 2. find the antiderivative of 4 - x, easy, x^2/2
step 3. plug in what the result of the...
Prove that:
\[ \int_{0}^{\pi/2} \cos(nx) \cos^n(x) dx =\frac{\pi}{2^{n+1}}\]
\[ \int_{0}^{\pi} \frac{1-\cos(nx)}{1-\cos(x)} dx =n\pi \]
where \( n \in \mathbb{N} \). You can use induction, contour integration or any other method you like.
By parameterizing the curve (not by Cauchy's theorem) and using the series of sin z, nd
the value of
∫z^k sin(z)dz around a closed Contour C where C is the unit circle z=e^(iθ), for 0≤θ<2π
What do they mean by using series of sin z ? I mean if I expand it .. I get e^(iθ)- e^(3iθ)/3! ---
and...
Homework Statement
Let f be a continuously differentiable function on the interval [0,2\pi], where f(0) = f(2\pi) and f'(0) = f'(2\pi). For n = 1,2,3,\dotsc, define
a_n = \frac{1}{2\pi} \int_0^{2\pi} f(x) \sin(nx) dx.
Prove that the series \sum_{n=1}^\infty |a_n|^2 converges...
Folks,
1) If we have \int F \cdot dr that is independent of the path, does that mean that the integral will always be 0?
2) For 2 dimensional problems when we evaluate line integrals directly and use Greens Theorem for every piece wise smooth closed curves C, arent we always calculating...
Homework Statement
Evaluate this integral directly
Homework Equations
\int cos x sin y dx +sin x cos y dy on vertices (0,0), (3,3) and (0,3) for a triangle
The Attempt at a Solution
Does this have to evaluated parametrically using r(t)=(1-t)r_0+tr_1 for 0 \le t\le 1
or can I just...
Integrals of Expeced Value For Normal Order Statistics
1. Find the expected value of the largest order statistic in a random sample of size 4 from the standard normal distribution.
Homework Equations
E(X(4,4))=4∫xf(x)(F(x))^3dx, (from minus infinite to plus infinite), where f(x) is the...
how would one evaluate this without using trig substitution? Is it possible to make one integral out of this?
{int[(y^2 + a1^2)^-1]dy +c1}/{int[(x^2 + a2^2)^-1]dx +c2} +c3
the numbers behind the 'a's and 'c's are supposed to be subscripts.
Also, how would one deal with this...
Hi,
I'm studying calculus 3 and am currently learning about conservative vector fields.
=============================
Fundamental Theorem for Line Integrals
=============================
Let F be a a continuous vector field on an open connected region R in ℝ^{2} (or D in ℝ^{3}). There exists...
Folks,
When we are evaluating integrals like the following, what are we evaluating in terms of units etc.
For example if I integrate Fdx I get an area which represents the energy where F is the force and d is the displacement so the units are Nm etc.
1) Integrals over intervals
...
Alright, I have a kind of dumb question:
Why do I distinguish between dq and dqi when considering the propagation from qi to q to qf?
For example, if we want the wave function at some qf and tf given qi and ti, we may write:
ψ(qf,tf)=∫K(qftf;qiti)ψ(qi,ti)dqi
Why do we distinguish between dqi...
In my PhD I need to solve an integral of the generalized MarcumQ function multiplied by a certain probability density function to get the overall event probability. Numerically solution produces bound result as it represents a probability but when trying to use a convergent power expansion of...
hi folks,
one often reads
\int_A f(x) dx = \int_A g(x) dx for arbirary A, thus f(x) = g(x), since the equaltiy of the Integrals holds for any domain A.
I don't see, why the argument "...for any domain A..." really justifies this conclusion.
Can someone explain this to me, please?
Alright, so I was wondering if anyone could help me figure out from one step to the next...
So we have defined |qt>=exp(iHt/\hbar)|q>
and we divide some interval up into pieces of duration τ
Then we consider
<q_{j+1}t_{j+1}|q_{j}t_{j}>
=<q_{j+1}|e-iHτ/\hbar|q_{j}>...
There are several improper integrals which keeps puzzling me. Let's talk about them in xoy plane. For simplicity purpose, I need to define r=sqrt(x^2+y^2). The integrals are ∫∫(1/r)dxdy, ∫∫(x/r^2)dxdy, ∫∫(x^2/r^4)dxdy, and ∫∫(x^3/r^6)dxdy. Here ‘^’ is power symbol. The integration area D...
Dear all,
I need some help with Gaussian. I would like to know if it is possible to make cube file of the density of the single molecular orbital in Gaussian (not just overall or alpha and beta spin density) and if it is possible, wuold you be so kind to tell me how to do that?
And also, is...
Homework Statement
Interpret the integrals (from 0 to 4)∫ (3x/4) dx + (from 4 to 5)∫ (sqrt(25-x^2)) dx as areas and use the result to express the sum above as one definite integral. Evaluate the new integral.
Homework Equations
The Attempt at a Solution
I see that I could...
Find the surface area of the triangle with vertices (0,0) (L, L) (L,-L)
I know I have to take the double integrals of f(x,y) but I have no idea what f(x,y) is supposed to be!
Homework Statement
Prove
\sqrt{\frac{2}{\pi}}\int^{\infty}_0x^{-\frac{1}{2}}\cos (xt)dx=t^{-\frac{1}{2}}
and use that to solve
\int^{\infty}_0\cos y^2dy
Is this good way to try to prove?
Homework Equations
The Attempt at a Solution
Homework Statement...
Homework Statement
http://img687.imageshack.us/img687/1158/skjermbilde20111204kl85.png
The Attempt at a SolutionI thought this was pretty hard and involved a number of different parts. Here's my work:
Let x=cosθ and z=sinθ, also let 0≤y≤2-x=2-cosθ. I parametrize Q1, which I define to be...
So we have 4 things:
-Scalar Line Integral
-integral of f(c(t))||c'(t)||dt from b to a
-length of C: integral on curve C of ||c'(t)||dt
-Vector Line Integral
-integral of F(c(t))●c'(t)dt from b to a
-Scalar Surface Integral
-surface integral: double integral of f(Φ(u,v))||n(u,v)||dudv on...
I am reading "Quantum Mechanics and Path Integrals" (by Feynman and Hibbs) and working out some of the problems... as a hobby of sorts.
I have run into a problem in section 12-6 Brownian Motion. On page 339 (of the emended edition), the authors demonstrate, by example, a method for...
derivative of integral over e^t to t^5 (sqrt(8+x^4)) dx
I know I need to use the chain rule and I can take the derivative of the integral without respect to e^t and t^5. If you know the answer, can you answer and tell me how to do it?! Calculus final on Monday...