Recent content by robertmatthew

Ah, I see it now. No idea why I kept thinking I could do that; I must have done this problem five or six times and did that every time. Thanks for your help!

Homework Statement Show that the function is a solution of the wave equation utt = a2uxx. u = (t)/(a2t2-x2) Homework Equations Quotient rule (f/g)' = (g*f ' - f*g') / g2 The Attempt at a Solution I began with the first and second partials of u with respect to x...

This is where I've been going wrong. In my last attempt, I sort of envisioned a box with rounded sides of the box but not rounded ends... so the same bubble-box but with flat ends... not really sure why I did that... So breaking up the shape, I count one whole cylinder that has the "height"...

I tried drawing it out the way you suggested. For a line segment, I got an oval around the segment that's one unit away from the line at every point. For a rectangle, I got an oval around it, leading me to think that for the 3D box it would be the ellipsoid. After looking at ...'s formula, I...

I'm confused then, if it's just a bubble, how can I find the volume?

Okay, so I looked up the volume of an ellipsoid and tried to work out the lengths of the axes. If the ellipsoid is 1 unit away from the box, then I figure the lengths of the semi-axes should be 1+.5L, 1+.5W, and 1+.5H, right? And then the volume is V=(4/3)(pi)(1+.5L)(1+.5W)(1+.5H)?

So rather than my initial thought of a sphere inside of the box, is the solution is a sphere outside the box where the distance from the sphere to say, the side of the box, is 1?

Homework Statement Consider a solid box with dimensions L,W, and H. Let S be the set of all points whose distance is at most 1 from the nearest point inside or on the box. What is the volume of S? Homework Equations Not sure if there are any? The Attempt at a Solution My initial...

Alright, I substituted u=x-R and solved far enough to find a shortcut using the area of a semicircle, and then evaluated to get 2π2Rr2. Thanks!

Would h be 2y, since the height of the cylinder depends on the curve of the torus? So for any point on the torus, where the circular cross section lies in the xy plane, y=sqrt(r2-(x-R)2). Then the cylindrical shell would dip above and below the x axis, so the height should be 2y, right...

Homework Statement Use cylindrical shells to find the volume of a torus with radii r and R. Homework Equations V= ∫[a,b] 2πxf(x)dx y= sqrt(r2 - (x-R)2) The Attempt at a Solution V= ∫ [R, R+r] 2πx sqrt(r2 - x2 - 2xR + R2) dx I feel like this isn't going in the right direction...

Are the other conditions f(0)=0 and f(x)≤x2? I don't understand how to verify those, would it have to do with the graph of the region? Since A=B and for A, both of the curves have a point at (0,0) so B must have two points there also? And how should I go about part b, do I just solve for f(x)?

Homework Statement Let a > 0 be a fixed real number. Define A to be the area bounded between y=x2,y=2x2, and y=a2. Define B to be the area between y = x2, y = f(x), and x = a where f(x) is an unknown function. a) Show that if f(0) = 0, f(x) ≤ x2, and A = B then int 0-->a2 [y1/2-(y/2)1/2] dy =...

I apologize in advance if the answer to this is really simple; I often overlook simple solutions when something trips me up. For example, if f(x)=x2 and g(x)=x3/2, and g(f(x)) is therefore, after simplification, x3, why is that still an even function if x3 graphed under other circumstances is...

Ah, okay. Got h^2+3h+3, entered it and it was correct. Thanks! I'm not sure why I thought it was asking for the limit, guess I'm still a little fuzzy from the summer.