http://calc3.askbot.com/questions/Open source question and answer forum written in Python and DjangoenCopyright Askbot, 2010-2011.Thu, 24 Jul 2014 21:30:49 -0500http://calc3.askbot.com/question/176/can-anyone-validate-problem-3/So for problem #3 on the In-Class worksheet I worked it through, but didn't have enough time to validate my answer with the rest of my group. I approached it as: $$\int_0^{2\pi} \int_0^{\sqrt{\pi/2}} \int_0^{\cos(r^2)} Z \delta z r \delta r \delta \theta$$ $$=2\pi \int_0^{\sqrt{\pi/2}} \int_0^{\cos(r^2)} Z \delta z r \delta r$$ $$=2\pi \int_0^{\sqrt{\pi/2}} \frac{\cos^2(r^2)}{2} r \delta r$$ And by the Double angle formula I got: $$=\frac{\pi}{2}\int_0^{\sqrt{\pi/2}}r(1+\cos(2r^2)\delta r$$ Using U-sub: $$u=2{r^2}$$ $$\delta u=4r \delta r$$ So I ended up with: $$=\frac{\pi}{8} \int_0^\pi 1+\cos(u) \delta u$$ $$=\frac{\pi}{8}(u+\sin(u)) |_0^\pi$$ Getting an answer of: $$\frac{\pi^2}{8}$$ If anyone got a different answer or can see an error in my math I would greatly appreciate your input.Gear JunkyThu, 24 Jul 2014 21:30:49 -0500http://calc3.askbot.com/question/176/http://calc3.askbot.com/question/165/graphical-explanation/Can someone give a graphical explanation (pictures please) as to how you can find the answer to question 1b of todays worksheet? $$\int_0^1\int_0^{\sqrt{1-x^2}}\int_0^{\sqrt{1-x^2-y^2}}1dzdydx$$ No one in my group could really grasp what McClure was saying in class.AnonymousThu, 24 Jul 2014 10:40:31 -0500http://calc3.askbot.com/question/165/http://calc3.askbot.com/question/163/setting-up-an-integral-for-the-volume-under-a-plane/I just want to know if I am doing this correctly. On the worksheet from today, number 6 asks: Set up an integral representing the volume under the plane $x+2y+z=2$ and in the first octant. I came up with the following: $$\int_0^2\int_0^\frac{2-x-z}{2}\int_0^{2-2y-z} dxdydz$$ am I doing this right or am I completely lost?AnonymousThu, 24 Jul 2014 10:14:06 -0500http://calc3.askbot.com/question/163/http://calc3.askbot.com/question/159/in-class-problem-5/I think that I understand how to set up the domain of this integral except for the inner integral. The question asks: "Let $D$ denote the set in $\mathbb{R^3}$ lying above the cone $z=\sqrt{x^2+y^2}$ and inside the sphere $x^2+y^2+z^2=4$. Set up the following integrals over $D$ as iterated integrals in spherical coordinates. You should think about which ones you can evaluate." would this be correct?: $$\int_0^\pi \int_0^\frac{\pi}{4} \int_0^2 (\rho^2)\rho^2\sin(\phi)d\rho d\phi d\Theta$$ comment: sorry, this is for part a.AnonymousThu, 24 Jul 2014 09:57:59 -0500http://calc3.askbot.com/question/159/http://calc3.askbot.com/question/158/in-class-problem/I am having trouble understanding #4e from todays worksheet: "Let $D$ denote the set in $\mathbb{R^3}$ lying above the cone $z=\sqrt{x^2+y^2}$ and below the plane $z=4$. Set up the following integrals over $D$ as iterated integrals in cylindrical coordinates. You should think about which ones you can evaluate. (e) An integral representing the volume of $D$." What is the significance of setting up an integral representing the volume of $D$? AnonymousThu, 24 Jul 2014 09:42:04 -0500http://calc3.askbot.com/question/158/http://calc3.askbot.com/question/149/in-class-problem/In class today my group and I had a little problem solving the second problem written on the board: Evaluate $\int\int\int sin((x^2+y^2+z^2)^{3/2}) dV$ where $D$ is defined as the top half of the solid unit sphere. This is how far we got $$\int_0^\pi\int_0^\pi\int_0^1 (sin((p^2)^{3/2})p^2 sin\phi dP d\phi d\Theta$$ $$\pi\int_0^\pi\int_0^1 sin(p^3)p^2 sin\phi dP d\phi$$ $$u=p^3$$ $$1/3 du=p^2 dP$$ $$\frac{\pi}{3}\int_0^\pi\int_0^1 sin(u) sin\phi du d\phi$$ $$\frac{\pi}{3}\int_0^\pi sin\phi - cos(u) \biggr|_0^1 d\phi$$ $$\frac{\pi}{3}\int_0^\pi 1-cos(1) d\phi$$ $$\frac{\pi}{3} (\phi-\phi cos(1)) \biggr|_0^\pi$$ $$\frac{\pi}{3} (\pi - \pi cos(1) $$ Did we start with the correct domain of integration or did we make a mistake in the calculations?AnonymousWed, 23 Jul 2014 09:41:37 -0500http://calc3.askbot.com/question/149/http://calc3.askbot.com/question/118/double-integrals/So with the problems he put on the board at the end of class today, I worked out the first double integral and was just hoping someone could verify if I'm doing it right. So he gave us the domain :  and the double integral as: $$\int\int x^2y\delta A$$ for which I came up with the integrals being: $$\int _0 ^2 \int _0 ^{4-x^2} x^2y \delta y \delta x$$ $$=\int _0 ^2 \frac 12x^2y^2\delta x \mid _0 ^{4-x^2}$$ $$=\int _0 ^2 \frac 12 x^2 (4-x^2)^2 \delta x$$ $$= \int _0 ^2 \frac 12 x^2(16-8x^2+x^4)\delta x$$ $$=\int _0 ^2 8x^2 -4x^4 + \frac 12 x^6 \delta x$$ $$=\frac 83 x^3 - \frac 45 x^5 + \frac {1}{14} x^7 \mid _0 ^2$$ $$= \frac 83 (8) - \frac 45(32) + \frac{1}{14}(128)$$ $$= \frac{64}{3} -\frac{128}{5} +\frac{128}{14}$$ This method seems to make sense to me(hoping that my terms of integration are correct), so I'm just hoping that someone else follows this too, or you could explain and easier way to do it. Thanks!TiffanyThu, 17 Jul 2014 10:21:11 -0500http://calc3.askbot.com/question/118/http://calc3.askbot.com/question/66/how-do-we-draw-the-left-and-right-contours-of-a-hyperbolic-function/I have been stumped on problem 3 for today's problem set, where we are to sketch a contour diagram of $f(x,y) = 4x^2 - 9y^2$. I know how to find the asymptotes (by setting $f(x,y) = 0$ and solving for $y$, which gives you asymptotes of $$y =\pm \frac{2}{3} x$$ which are very easy to plot). I can then solve for the "top and bottom" hyperbola by setting $f(x,y) = 1$, and solving for $y$ which gives "top and bottom" contours of $$y =\pm \sqrt( \frac{2}{3} x^2 -1)$$ which I can easily plot. This is where I have problems. How do I plot the left and right hyperbola? What would I set the function to so as to do this? I apologize for the lack of graphs, I wasn't sure how to plot them to show. If you have questions about what I mean for "left and right" contours, let me know. Thanks!WesTue, 08 Jul 2014 21:11:10 -0500http://calc3.askbot.com/question/66/http://calc3.askbot.com/question/19/the-bug-on-the-wheel-goes-round-and-round/I'm pretty confident that I understand how to parametrize the bug's movement. But I don't understand what kind of movement I need to parametrize. can someone clarify the bugs movement in non-mathematical terms. Is the wheel moving in any dimension or just spinning? SpaceManSpiffMon, 30 Jun 2014 09:26:18 -0500http://calc3.askbot.com/question/19/http://calc3.askbot.com/question/36/sphereswill-this-workin-class-3/So if you set the equations for the two spheres equal to each other you get a plane. $$x^{2}+y^{2}+z^{2}=(x-1)^{2}+(y-1)^2+(z-1)^{2}$$ $$-2x-2y-2z=3$$ I believe that simplification is correct. The question is, if I parametrize the circle created by the intersection of the unit sphere with this plane, will that describe the intersection of the two spheres?SpaceManSpiffTue, 01 Jul 2014 09:43:05 -0500http://calc3.askbot.com/question/36/