1. ### Verifying the flux transport theorem

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...
2. ### How to prove that ##f(x,y)## is not integrable over a square?

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...
3. ### Potential energy of a shell and a disc, both covered uniformly with charge

Double integration maybe?? I calculated potential due to shell on plate's center but not on other points on it's surface.
4. ### I Change of order in double integrals

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...
5. ### A Maximization Problem

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...
6. M

### Using Green's Theorem for a quadrilateral

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...
7. H

### Faraday's law with Calc 3 integration help

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...
8. D

### Trying to find this double integral using polar coordinates

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...
9. ### Integrating with respect to area? Past paper question

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...
10. ### Double integral, find volume of solid

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...
11. ### Issue with Double Integral

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...
12. ### I Integration Limits Changing in Double Integral Order Change

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...
13. ### Moment of inertia (double integral)

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...
14. ### Double integral solution

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 :(
15. ### Calc III Double Integral Question

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...
16. ### Surface integrals

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...
17. ### Problem integrating a double integral

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...
18. ### The quarter disk in the first quadrant bounded by x^2+y^2=4

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...
19. ### Volume bounded by two surfaces, what am I missing?

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...
20. ### Force on a superconducting cube

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...
21. ### Gravity of a disk acting on a mass on the z axis

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...
22. ### Multiple Integral Challenge Question, no solution in guide

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...
23. ### Double Integrals: Where am I making a mistake?

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...
24. ### Calculating Flux through Ellipsoid

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 ##...
25. ### Leibniz rule for double integrals

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...
26. ### Two variable function, single integral

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...
27. ### Hard integrals in 2D

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)...