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...
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...
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...
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
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
Suppose an infinitely long wire carrying current ##I=sin_0(\omega t)## is a distance ##a## away from a equilateral triangular circuit with resistance ##R## in the same plane as shown in the figure. Each side of the circuit is length ##b##. I need to find the induced voltage...
Homework Statement
question :
find the value of
\iint_D \frac{x}{(x^2 + y^2)}dxdy
domain : 0≤x≤1,x2≤y≤x
Homework Equations
The Attempt at a Solution
so here, i tried to draw it first and i got that the domain is region in first quadrant bounded by y=x2 and y=x
and i decided to...
This isn't exactly homework or coursework, it is a past paper question that I cannot find a solution to (my university doesn't like releasing answers for some reason unknown to me).
The question is attached as an image (edit: the image displays while editing but not in the post, so I'll try to...
Homework Statement
Find the volume of the solid by subtracting two volumes, the solid enclosed by the parabolic cylinders:
y = 1 − x2,
y = x2 − 1
and the planes:
x + y + z = 2
4x + 5y − z + 20 = 0
Homework Equations
∫∫f(x,y) dA
The Attempt at a Solution
So I solved for z in the plane...
Homework Statement
Find the volume of the given solid:
Under the surface z = xy and above the triangle with verticies (1,1), (4,1) and (1,.2)
Homework Equations
Double Integral
The Attempt at a Solution
I drew the triangle, and found the the equations of the lines to be:
x = 1;
y = 1;
y = -3x...
For part of a proof of a differential equations equivalence, we needed to use that $$\int_0^t [\int_0^s g(\tau,\phi(\tau))\space d\tau]\space ds = \int_0^t [\int_\tau^t ds]\space g(\tau,\phi(\tau))\space d\tau$$
I understand that the order is being changed to integrate with respect to s first...
Homework Statement
Determine the moment of inertia of the shaded area about the x axis.[/B]
Homework Equations
Ix=y^2dA
The Attempt at a Solution
Okey so I now get how to do this the standard method. But I want to know if the method I tried is correct aswell or where my mistake lies.
My...
Homework Statement
##\int_{z=0}^5 \int_{x=0}^4 \Big( \frac{xz}{ \sqrt{16-x^2}} +x \Big)dxdz##
Homework Equations
double integration
The Attempt at a Solution
how do i integrate the term ##\frac{xz}{ \sqrt{16-x^2}}## though i know that ##\int x \, dx = \frac{x^2}{2}##
pls help me thoroughly :(
This is the problem I'm trying to solve. The directions require me to rewrite as a single integral and evaluate. But I'm having trouble setting the bounds for a complete compounded integral. The graph of the region would look something like this...
Where the shaded area is the region. I...
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...
Hi, could you please help with the integration of this equation:
$$\int_{x}\int_{y}\frac{\partial}{\partial y}\left(\frac{\partial u}{\partial x}\right)\,dydx$$
where ##u(x,y)## . From what I remember, you first perform the inner integral i.e. ##\int_{y}\frac{\partial}{\partial...
Find the coordinate of center of mass.
Given: The quarter disk in the first quadrant bounded by x^2+y^2=4
I tried to solve this problem but can't figure out how to do it.
so y integration limit is: 0 <= y <= sqrt(4-x^2))
x limit of integration: 0 <= x <= 2
and then after the dy integral I...
Homework Statement
Find the volume of the solid bounded by z=x^2+y^2 and z=8-x^2-y^2
Homework Equations
use double integral dydx
the text book divided the volume into 4 parts,
The Attempt at a Solution
[/B]
f(x)= 8-x^2-y^2-(x^2+y^2)= 4-x^2-y^2
i use wolfram and got 8 pi, the...
Hi everyone,
I need some help to look if I did these calculations right.
Let us assume a three dimensional magnetic field:
##\vec{B}(x,y,z) = B_x(x,y,z)\hat{x} + B_y(x,y,z)\hat{y} + B_z(x,y,z)\hat{z}##
The equation for the force on a superconducting particle in a magnetic field is given by...
Homework Statement
A lamina has constant density \rho and takes the shape of a disk with center the origin and radius R. Use Newton's Law of Gravitation to show that the magnitude of the force of attraction that the lamina exerts on a body of mass m located at the point (0,0,d) on the positive...
I have what I think is a valid solution, but I'm not sure, and when I try to check the answer approximately in Matlab, I don't get a verified value, and I'm not sure if my analytic solution or my approximation method in Matlab is at fault.
1. Homework Statement
Evaluate the integral...
Homework Statement
Find the volume of the solid.
Under the paraboloid z = x^2 + y^2 and above the region bounded by y = x^2 and x = y^2
Well, those curves only intersects in the xy-plane at (0,0) and (1,1), and in the first Quadrant, and in that first Quadrant y = sqrt(x), and over that...
Homework Statement
Let ## E ## be the ellipsoid:
$$ \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+z^{2}=1 $$
Let ## S ## be the part of the surface of ## E ## defined by:
$$ 0 \leq x \leq 1, \ 0 \leq y \leq 1, \ z > 0 $$
Let F be the vector field defined by $$ F=(-y,x,0)$$
A) Explain why ##...
Hello, I would like to differentiate the following expected value function with respect to parameter $$\beta$$: $$F(\xi_1,\xi_2) =\int_{(1-\beta)c_q}^{bK+(1-\beta)c_q}\int_{(1-\beta)c_q}^{bK+(1-\beta)c_q}\frac{\xi_1+\xi_2-2bK}{2(1-\beta)^2} g(\xi_1,\xi_2)d\xi_1 d\xi_2$$ $$g(\xi_1,\xi_2)$$ is...
Homework Statement
Evaluate:
I(y)= \int^{\frac{\pi}{2}}_{0} \frac{1}{y+cos(x)} \ dx if y > 1
Homework Equations
The Attempt at a Solution
I've never seen an integral like this before. I can see it has the form:
\int^{a}_{b} f(x,y) dx
I clearly can't treat it as one half of an exact...
I can't compute the integral:
\int \frac{\arccos(\sqrt{x^2+y^2})}{\sqrt{x^2+y^2}}\frac{x-a}/{(\sqrt{(x-1)^2+y^2})^3 dxdy
on an unit circle: r < 1.
for const: a = 0.01, 0.02, ect. up to 1 or 2.
I used a polar coordinates, but the values jump dramatically in some places (around the 'a' values)...