# Search results

1. ### First Order Linear DE System

thanks, I figured it out
2. ### First Order Linear DE System

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...
3. ### Jacobian Change of Variables Question

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...
4. ### Jacobian Change of Variables Question

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...
5. ### Spherical Coordinates Integral

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...
6. ### Spherical Coordinates Integral

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...
7. ### Spherical Coordinates Integral

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?
8. ### Spherical Coordinates Integral

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...
9. ### Polar Coordinates Improper Integral Proofs

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...
10. ### Polar Coordinates Improper Integral Proofs

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...
11. ### Curvature Proof

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?
12. ### Vector Application Problem

Oh wait. you know v(0)=0. So this could be arranged to be m(t)\frac{1}{e^2}=m(0) so it uses 1/e2 of it's fuel. Is this what you were thinking?
13. ### Vector Application Problem

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...
14. ### Proof of r(t) and r'(t) orthogonal on a sphere

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...
15. ### Proof of r(t) and r'(t) orthogonal on a sphere

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...
16. ### Proof of r(t) and r'(t) orthogonal on a sphere

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...
17. ### Surface with specifications

Thanks. You wind up with the cone 4y^2+4z^2=x^2
18. ### Surface with specifications

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...
19. ### Tricky Calculus III Problem

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...
20. ### Tricky Calculus III Problem

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...
21. ### Integration By Parts Problem

Never mind, I've found my stupid mistake. Distributive Property!
22. ### Integration By Parts Problem

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
23. ### Sketching A Graph Using Differentiation

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...
24. ### Difference Quotient Isn't working.

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.
25. ### Difference Quotient Isn't working.

\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...
26. ### Difference Quotient Isn't working.

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...