What is Double integral: Definition and 573 Discussions
In mathematics (specifically multivariable calculus), a multiple integral is a definite integral of a function of several real variables, for instance, f(x, y) or f(x, y, z). Integrals of a function of two variables over a region in
R
2
{\displaystyle \mathbb {R} ^{2}}
(the real-number plane) are called double integrals, and integrals of a function of three variables over a region in
R
3
{\displaystyle \mathbb {R} ^{3}}
(real-number 3D space) are called triple integrals. For multiple integrals of a single-variable function, see the Cauchy formula for repeated integration.
$$h(t)=f(t)*g(t)=\int_0^t f(\tau)g(t-\tau)d\tau=\int_0^t g(\tau)f(t-\tau)d\tau\tag{1}$$
The Laplace transform is
$$H(s)=\int_0^\infty h(t)e^{-st}dt=\int_0^\infty\left ( \int_0^t g(\tau)f(t-\tau)d\tau\right )e^{-st}dt\tag{2}$$
The Laplace transforms of $f$ and $g$ are
$$F(s)=\int_0^\infty...
Ok in my approach i have the lines,
starting with the inner integral,
$$\int_0^1 xy \cos (x^2y) dx$$
I let ##u =x^2y , u(0)=0, u(1)=y##
...
$$\dfrac{1}{2} \int_0^y \cos u du=\left[\dfrac{1}{2} \sin u \right]_0^y= \left[\dfrac{1}{2} \sin (x^2y) \right]_0^1=\left[\dfrac{1}{2} \sin y...
This homework statement comes from a research paper that was published in SPIE Optical Engineering. The integral $$\int\int_{-\infty}^{\infty}drdr'W(\vec{r})W(\vec{r'}) \vec{r} \cdot \vec{r'}=0$$ is an assumtion that is made via the following statement from the paper : "Since...
I am trying to do the double integral.
And I remembered there's this formula that says if the integrand can be split into products of F(x) and G(y) then we can do each one separately, then take the product of each result. Taken from Stewart's Calculus 9E.
So I tried to do the integral two...
I have the following problem and am almost sure of the answer but can't quite prove it:
##f(y)## is nonnegative, and I know that ##\int_0^{\infty } f(y) \, dy## is finite.
I now need to calculate (or simplify) the double integral:
$$\int_0^{\infty } \left(\int_x^{\infty } f(y) \, dy\right) \...
Question: Suppose I have a data file for the acceleration of an object after every ##
\Delta t_i##, how do I obtain the displacement of it?
Context: Integral in a PID loop, although not exactly what I am asking as one is sum of error: $$\int_0^T \int_0^T \ddot {\vec \theta(t)}dtdt$$
the other...
So I want to subtract the two surfaces, right? I really don't know where to start... I am guessing this would be some sort of triple integral, however I am very confused with the bounds.
Any help would be greatly appreciated!
Thanks!!
The implicit curve in question is ##y=\operatorname{arccoth}\left(\sec\left(x\right)+xy\right)##; a portion of the equations graph can be seen below:
In particular, I'm interested in the area bound by the curve, the ##x##-axis and the ##y##-axis. As such, we can restrict the domain to ##[0...
Write a program that uses the Monte Carlo method to approximate the double integral $\displaystyle\iint\limits_R e^{xy}dA$ where $R = [-1,1] \times [0, x^2]$. Show the program output for N = 10, 100, 1000, 10000, 100000 and 1000000 random points.
My correct answer:
My Java program...
I have the solution for this problem using dydx as the area. Worse yet, I cannot find another solution for it. Everyone seems to just magically pick dydx without thinking and naturally this is frustrating as learning the correct choice is 99.9% of the battle...
So, I was curious how one might...
I need help with a derivation of an equation given in a journal paper. My question is related to the third paragraph of this paper: https://doi.org/10.1007/BF00619826. Although it is about fibre coupling my problem is purely mathematical. It is about solving a complex double integral. The...
Greetings All!
I have a problem finding the correct solution at first glance
My error was to determine the region of integration , for doing so I had to the intersection between y= sqrt(x) and y=2-x
to do so
x=(2-x)^2
to find at the end that x=1 or x=5
while graphically we can see that the...
Greetings!
I have the following integral
and here is the solution of the book (which I understand perfectly)
I have an altenative method I want to apply that does not seems to gives me the final resultMy method
which doesn't give me the final results!
where is my mistake?
thank you!
I sketched this out. With the z=0 and y=0 boundaries, we are looking at ##z \geq 0## and ##y \geq 0##
I believe ##0 \leq x \leq 5## because of the boundary of ##y=\sqrt{25-x^2}##.
This is my region
##\int_0^5 \int_0^\sqrt{25-x^2} x \, dydx ##
## =\int_0^5 xy \vert_{0}^{\sqrt{25-x^2}} \, dx##...
Summary:: Could someone please evaluate this double integral over rectangular bounds? Answer only is just fine.
[Mentor Note -- thread moved from the technical math forums, so no Homework template is shown]
Hi,
I'm trying to find the answer to the following integral over the rectangle...
Background information
Earlier they've shown that some double integrals can be simulated if it contains pdfs.
Ex: $$\int \int cos(xy)e^{-x-y^2} dx dy$$
Can be solved by setting:
Exponential distribution
$$f(x) = e^{-x}, Exp(1)$$
Normal distribution
$$f(y) = e^{-y^2}, N(0, 1/\sqrt 2)$$By knowing...
The answer calculates the integral with ##du## before ##dv## as shown below.
However I decided to compute it in the opposite order with different bounds. Here is my work:
According to the definitions, $$\begin{cases} u=x+y\\ v=2x-3y \end{cases}$$
First we need to convert the boundaries in xy...
Summary:: Calculate a double integral via appropriate change of variables in R^2
Suppose I have f(x,y)=sqrt(y^12 + 1). I need to integrate y from (x)^(1/11) to 1 and x from 0 to 1. The inner integral is in y and outer in x. How do I calculate integration(f(x,y)dxdy) ?
My Approach: I know that...
Dear all,
Last semester on the final exam, our professor gave us an integral that seems difficult to solve.
The integral came at the end of a lengthy problem, where we were asked to find the net Gauss curvature of Enneper's surface.
The integral that emerged is the following.
We tried...
Performing the x-integration first the limit are x=y2 and x= -y2 and then the y limits are 0 to 1. This gives the final answer 2/5
But i am getting confused when trying to reverse the order of integration. My attempt is that i have to divide the region in 2 equal halfs and then double my answer...
I know the value of this integral is equal to 0, but I would like to see if there is any tricks to spot this answer using symmetries or even odd propreties?
Thanks in advance
calculate the double integral
over the region of integration is
x^2 + y^2 ≤ 4; x^2 + (y/4)^2 ≥ 1
the integrals have been made over two regions
my problem is that when I go to the polar coordinate for the ellipsis and use the jacobian i got 2 instead of 8 ( the following is the professor...
I already have the solution in which the region of integration has been divided into two regions
but I was wondering if I can only use one region considering the polar coordinate system) the disk equation for me is r=2cos(θ) and the theta goes from 0 to (pi/4)
0<r<2cos(θ) and the 0 <θ<pi/4...
Let ##S_t## be a uniformly expanding hemisphere described by ##x^2+y^2+z^2=(vt)^2, (z\ge0)##
I assume by verify they just want me to calculate this for the surface. I guess that ##\textbf{v}=(x/t,y/t,z/t)## because ##v=\frac{\sqrt{x^2+y^2+z^2}}{t}##. The three terms in the parentheses evaluate...
are the boundaries of integration correct?
i split the domain in two as follows
-2<=x<=0 , -(4-x^2)^(1/2)<y<=x+2 and
0<=x<=2 -(4-x^2)^(1/2)<=y<=(4-x^2)^(1/2)
D={(x,y)∈ℝ2: 2|y|-2≤|x|≤½|y|+1}
I am struggling on finding the domain of such function
my attempt :
first system
\begin{cases}
x≥2y-2\\
-x≥2y-2\\
x≥-2y-2\\
-x≥-2y-2
\end{cases}
second system
\begin{cases}
x≤y/2+1\\
x≤-y/2+1\\
-x≤y/2+1\\
-x≤-y/2+1\\
\end{cases}
i draw the graph and get the...
Hi everyone, I was wondering if it was possible to calculate a double integral by converting it to a line integral, using the greens theorem, and if so is it possible to get a non zero answer. if we were working on a rectangular region
Hi everyone, I tried to solve the last part of the question, I substituted back the expression of x and y into the equation of the ellipse, I got that r=1 or r=-1. But got no idea how to find the boundary for theta, I got a guess that, It should be from zero to pi. But got no reason why to...
Hello there,
I'm struggling in this problem because i think i can't find the right ##\theta## or ##r##
Here's my work:
##\pi/4\leq\theta\leq\pi/2##
and
##0\leq r\leq 2\sin\theta##
So the integral would be: ##\int_{\pi/4}^{\pi/2}\int_{0}^{2\sin\theta}\sin\theta dr d\theta##
Which is equal to...
I have an integral:
\int_{-1}^{0}\int_{-1}^{q}\delta(s+a)\sinh[k(q-s)]dsdq
where 0<a<1 and \delta (s-a) is a dirac delta function. Anyone know what to do?
Hi I´d like a suggestion about a surface double integral. If I have a sphere x^2+y^2+z^2=4 is on the top of a cardioid r=1-cosθ. The problem is when I solve the integral I got a inverse sine when the answer is a natural logarithm (ln)
I'm confused with how Riemann sums work on double integrals. I know that ##L=\sum_{i,j}fm_{ij}A_{ij}## and ##U=\sum_{i,j}fM_{ij}A_{ij}## where ##m_{ij}## is the greatest lower bound and ##M_{ij}## is the least uper bound and ##A_{ij}## is the area of each partition.
##A_{ij}=\frac{1}{n^2}## for...
So i drew sketch.
And I do not understand, how to write integral for calculation, which I should use, X or Y on limit?
Is one of them right?
First answer gives me 65,7
Second 383,4
Homework Statement
Integrate from 0 to 1 (outside) and y to sqrt(2-y^2) for the function 8(x+y) dx dy.
I am having difficulty finding the bounds for theta and r.
Homework Equations
I understand that somewhere here, I should be changing to
x = r cost
y = r sin t
I understand that I can solve...
In the question given below, can we change the order of integral so that y can be the independent variable and x be the dependent one?The cylinder x^2 + z^2 = 1 is cut by the planes y=0,z=0 and x=y.Find the volume of the region in the first octant.This may look like a homework question but it's...
Homework Statement
Evaluate ##\int\int_{R} (x+2)(y+1) \; dx \; dy## where ##R## is the pentagon with vertices ##(\pm 1,0)##, ##(\pm 2,1)## and ##(0,2)##.
Homework EquationsThe Attempt at a Solution
After drawing ##R## I split ##R## into two sections ##R_1## (left half) and ##R_2## (right half)...
I have a double integral where the region of integration is ##R = \{(u,v) : 0 \le u < \infty, ~0 \le v < \infty) \}##. I am doing the change of variables ##u=zt## and ##v = z(1-t)##. I am a bit rusty on calc III material, so how would I find the new region of integration, in terms of the...
Homework Statement
Determine the area of the surface A of that portion of the paraboloid:
[x][/2]+[y][/2] -2z = 0
where [x][/2]+[y][/2]≤ 8 and y≥x
Homework Equations
Area A = ∫∫ dS
The Attempt at a Solution
Area A = ∫∫ dS = 3∫∫ dS
$\tiny{232.5a}\\
\textsf{Evaluate the double integral}$
\begin{align*}\displaystyle
I_a&=\iint\limits_{R} xy\sqrt{x^2+y^2} \, dA \\
R&=[0,2]\times[-1,1]
\end{align*}
Ok, just want to see if I made the first step correct.
this looks like simply a rectangle so x and y are basically...
Consider a double integral
$$K= \int_{-a}^a \int_{-b}^b \frac{B}{r_1(y,z)r_2^2(y,z)} \sin(kr_1+kr_2) \,dy\,dz$$
where
$$r_1 =\sqrt{A^2+y^2+z^2}$$
$$r_2=\sqrt{B^2+(C-y)^2+z^2} $$
Now consider a function:
$$C = C(a,b,k,A,B)$$
I want to find the function C such that K is maximized. In other...
Homework Statement
r=1 and r=1+cos(theta), use a double integral to find the area inside the circle r=1 and outside the cardioid r=1+cos(theta)
Homework EquationsThe Attempt at a Solution
I am confused on the wording and how to set it up. I tried setting it up by setting theta 0 to pi. and r...
Evaluate (use attached figure for depiction) $ \iint_{R} \, xy \, dA $
where $R$ is the region bounded by the line
$y = x - 1$ and the parabola $y^2 = 2 x + 6$.
I will post solution in just a moment with a reply.
Homework Statement
Evaluate the line integral of (sin x + y) dx + (3x + y) dy on the path connecting A(0, 0) to B(2, 2) to C(2, 4) to D(0, 6). A sketch will be useful.
Homework Equations
Sketching the points, I have created a parallelogram shape. I also know that green's theorem formula, given...
Homework Statement
z=x^2+xy ,y=3x-x^2,y=x find the volume of the region
Homework EquationsThe Attempt at a Solution
I graphed y=3x-x^2 and y=x I am confused on which region I use to find the volume. Do I use the upper region or the lower region.
Homework Statement
I'm given the integral show in the adjunct picture, in the same one there is my attempt at a solution.
Homework Equations
x = r.cos(Θ)
y = r.sin(Θ)
dA = r.dr.dΘ
The Attempt at a Solution
[/B]
I tried to do it in polar coordinates, so I substituted x=r.cos(Θ) y=r.sin(Θ) in...