Homework Statement
I need to solve this DE system for a lab:
q_1'=2-\frac{6}{5}q_1+q_2
q_2'=3+\frac{3}{5}q_1-\frac{3}{2}q_2
Homework Equations
The Attempt at a Solution
I know how to use the method of elimination to solve such systems, but this is non homogeneous because of the added...
you can rewrite the exponents as (x-y)(x+y). Using the substitution u=x-y and y=x+y you have a parallelogram with bounds v=3u,v=3u-8,v=-2u+1,v=-2u+8. Therefore, the area can be represented by the following integrals...
Homework Statement
Evaulate the integral making an appropriate change of variables.
\int\int_R(x+y)e^{x^2-y^2}dA where R is the parallelogram enclosed by the lines x-2y=0, x-2y=4, 3x-y=1, 3x-y=8 .
Homework Equations
The Attempt at a Solution
I'm not sure what change of variables I should...
Hmm, that's what I thought you were thinking. I handed in the assignment today and got the answer key. They setup the integral the way you did in the answer key. However, I would argue that part of the region in that integral is not under the cone. Everything under the cone is contained within a...
Ok, I think I was misinterperting the volume described. I took it to mean basically the shadow cast down by the cone on the xy-plane, since that region lies under the cone, above the xy-plane, and in the sphere. I'm not sure if the area you are thinking of is correct though. In the area you are...
Oh, haha. Cant believe I forgot that. I would think the upper limit on rho depends on phi because with this shape, you can change theta all you want and rho is not going to change. However, changing phi changes rho, this is because the solid is symmetrical about the z-axis. Right?
Homework Statement
Using spherical coordinates, find the volume of the solid that lies within the sphere x2+y2+z2=4, above the xy-plane and below the cone z=√(x2+y2)
Homework Equations
The Attempt at a Solution
This is what I have so far...
update, I proved everything, however I'm not sure If my proof for b is what they're asking for. I said that
\int_{-\infty}^{\infty}e^{-x^2}dx\int_{-\infty}^{\infty}e^{-y^2}dy=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-(x^2+y^2)}dy dx
and since I had proved that the rhs of the...
Homework Statement
(a) we define the improper integral (over the entire plane R2)
I=\int\int_{R^2}e^{-(x^2+y^2)}dA=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-(x^2+y^2)}dy dx=\lim_{a\rightarrow\infty}\int\int_{D_{a}} e^{-(x^2+y^2)} dA
where Da is the disk with radius a and center the...
Homework Statement
show that the curvature of a plane curve is \kappa=|\frac{d\phi}{ds}| where phi is the angle between T and i; that is, phi is the inclination of the tangent line.
Homework Equations
The Attempt at a Solution
I'm not sure how to start this one out.
Any ideas?
Homework Statement
A rocket burning it's onboard fuel while moving through space has a velocity v(t) and mass m(t) at time t. If the exhaust gasses escape with velocity ve relative to the rocket , it can be deduced from Newton's Second Law of Motion that...
Ok, I stared at the problem for a good amount of time and I think I have it. I managed to manipulate a few things...
2( \vec{r}(t) \cdot \vec{r}'(t))=\frac{d}{dt}(\vec{r}(t)\cdot\vec{r}(t))=\frac{d}{dt}(\left\|\vec{r}(t)\right\|^2)=0
so it must be true then that also...
Hmm.
Lineintegral1:
Ok, I can rewrite what I have written.
\vec{r}(t) \cdot \vec{r}'(t)=0
\left\|\vec{r}(t) \right\|\left\|\vec{r}'(t)\right\|cos\theta=0
||r(t)|| has to equal the radius of the sphere. However, isen't that what I'm suppose to be proving given that r(t) and r'(t) are...
Homework Statement
if a curve has the property that the position vector r(t) is always perpendicular to the tangent vector r'(t). show that the curve lies on a sphere with center the origin.
Homework Equations
The Attempt at a Solution
I'm not quite sure how to prove this.
I...
Homework Statement
Find an equation for the surface consisting of all points p for which the distance from P to the x-axis is twice the distance from P to the yz-plane. Identify the surface.
Homework Equations
The Attempt at a Solution...
Genius! I love it. Can't believe I didn't think of that.
so for the line I I created the equation x=-2t,y=t,z=-2t
Next I plugged the equation into the sphere with it's center at (-2,1,-2) obtaining...
(-2t+2)^2+(t-1)^2+(-2t+2)^2=4
2(4t^2-8t+4)+(t^2-2t+1)=4
9t^2-18t+5=0...
Homework Statement
Find the volume of the solid that lies in between both of the spheres:
x2+y2+z2+4x+2y+4z+5=0
and
x2+y2+z2=4
Homework Equations
This is the first chapter of the calculus III material so no double or triple integrals are needed to solve this problem.
The Attempt at a...
Homework Statement
Homework Equations
The Attempt at a Solution
I tried this problem and couldn't figure it out so I went and got the solution. However, I don't understand step 6 of the solution. I'm not sure how
(n-1)\int\sin^{n-2}x(1-\sin^2x)dx=(n-1)\int\sin^{n-2}dx-(n-1)\int\sin^nx dx
Homework Statement
Homework Equations
The Attempt at a Solution
I don't understand how they get from step 4 to step 5. Wouldn't you factor a cos x out of the brackets then have (\cos x)(1-\frac{1}{6}) to the left of the brackets. Then you can multiply the (1-\frac{1}{6}) inside...
Nevermind you would simply factor out a delta x out of the numerator then you have 2 'delta x's that cancel out. Then you take the limit and you get 9x2-9. Otherwise its in indiscriminate form.
\lim_{\Delta x\rightarrow0}=\frac{9x^2\Delta x+9x\Delta x^2+3\Delta x^3-9\Delta x}{\Delta x}
Ok, so you would get this after taking the limit.
=9x^2+9x\Delta x+3\Delta x^2-9
what happens to the delta x terms though? Is delta x just such a small number \epsilon that you can say they are...
Homework Statement
Find the derivative of f using the differance quotient and use the derivative of f to determine any points on the graph of f where the tangent line is horizontal.
f(x)=3x^3-9x
Homework Equations
The Attempt at a Solution
\lim_{\Delta...