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    Triple Integral of Tetrahedron

    Homework Statement Evaluate the triple integral \int\int\int^{}_{E} xy dV where E is the tetrahedron (0,0,0),(3,0,0),(0,5,0),(0,0,6). Is there a simple way to simplify the integration? Homework Equations The Attempt at a Solution \frac{z}{6} + \frac{y}{5} + \frac{x}{3} = 1 z =...
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    Triple Integrals

    I guess I shouldn't have rushed through the calculations as I did, thanks a lot for pointing out my mistake.
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    Triple Integrals

    Didn't even think about that . . .but for some reason it doesn't seem to be working. I switched the integration around and got the following. \int_0^1 \int_0^4 5ye^{-xy} dxdy After the first integration it comes out really neatly as -5e-xy and after plugging in 0 and 4 I get -5e-4y...
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    Triple Integrals

    Homework Statement \int\int\int^{}_{B} ye^(-xy) dV where B is the box determined by 0 \leq x \leq 4, 0 \leq y \leq 1, 0 \leq z \leq 5. Homework Equations The Attempt at a Solution \int^{4}_{0}\int^{1}_{0}\int^{5}_{0} ye^(-xy) dzdydx Integrating the first time I get zye-xy Plugging in 5...
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    Double Integral in Polar Coordinates

    Okay, then after integration I get the following. 128*pi/2 - [1/2\theta + sin(2*\theta)/4] Plugging in my values I get the following. 128*pi/2 - pi/4 - sin(pi)/4 + sin(0)/4 but this is wrong. I am not sure what I am doing wrong.
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    Double Integral in Polar Coordinates

    That seems to make sense since if I think of \theta as how much an imaginary line sweeps through and radius as how long that line actually is. So now that I have the following: \int^{\pi/2}_{0}\int^{16}_{0} r drd\theta - \int^{\pi/2}_{0}\int^{16cos\theta}_{0} r drd\theta After the first...
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    Double Integral in Polar Coordinates

    I sketched it out and for the second one I am basing my intervals of integration of r from the center of the circle to the edge of the circle.That seems to make the most sense to me, however, a similar example in my book (which has a circle shifted 2 to the right instead of 8 and has a radius of...
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    Double Integral in Polar Coordinates

    Homework Statement Using polar coordinates, evaluate the integral which gives the area which lies in the first quadrant between the circles x2 + y2 = 256 and x2 - 16x + y2 = 0. Homework Equations The Attempt at a Solution Finding the intervals of integration for the polar coordinates...
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    Change of Variables

    Oh I am so sorry about that, I completely mistyped that. It was supposed to be dx/du, dx/dv, dy/du, dy/dv. Also thanks you pointed out where I got it wrong. That helped a lot.
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    Change of Variables

    I think I calculated my Jacobian correctly since I am taking the correct partial derivatives of my functions. dx/du -1/7u + 3/7v = -1/7 dx/dv -1/7u + 3/7v = 3/7 dy/du 2/7u - 3/7v = 2/7 dy/dv 2/7u - 3/7v = 3/7 -1/7*2*7-3/7*3/7 = -2/49-9/49 = -11/49 and absolute value gives me 11/49
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    Change of Variables

    Homework Statement Suppose D is the parallelogram enclosed by the lines 2x-3y = 0, 2x-3y = 2, 3x-y = 0 and 3x-y = 1. \int\int^{}_{D} [(2x-3y) e^(3x-y) dA Homework Equations The Attempt at a Solution Set u to be equal to 2x-3y -> x = (u+3y)/2 Set v to be equal to 3x-y -> v = 3/2(u+3y)-y...
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    Change of Variables

    Homework Statement Suppose D is the parallelogram in the xy-plane with vertices P(-1,5), Q(1,-5), R(5,-1), S(3,9) \int\int ^{}_{D} (6x+12y) dA HINT: Use transformation x = \frac{1}{6}(u+v) and y = \frac{1}{6} (-5u+v). Homework Equations The Attempt at a Solution Calculating the Jacobian I...
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    Double Integral

    It was cos(x^2), and I think I got it using integration by substitution. Thanks a lot.
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    Double Integral

    Homework Statement Evaluate the integral by reversing the order of integration. \int^{3}_{0}\int^{9}_{y^2} y cos(x^2) dydx Homework Equations ...? The Attempt at a Solution Drawing the picture out we get a sideways parabola. From the picture I get the following intervals of integration...
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    Volume of Solid

    Homework Statement Find the volume of the solid bounded by the planes x = 0, y = 0, z = 0, and x + y + z = 9. Homework Equations . . . ? The Attempt at a Solution After drawing out the picture with z=0 I have a line going from 0,9 to 9,0 bounded by the x and y axis giving me a triangle...
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    Iterated Integral

    You, were right! I screwed up by substituting the limits into the y instead of x by mistake. Thanks a lot.
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    Iterated Integral

    Homework Statement Evaluate the iterated integral I = \int^{1}_{0}\int^{1+y}_{1-y} (6y^2+ 10x) dxdy Homework Equations . . . ? The Attempt at a Solution Integrate with respect to x gives me the following equation. \int^{1}_{0} 6xy^2 + 5x^2 dy I plug in y+1 and y-1 into x and get the...
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    Double Integrals

    Oh, I see what I did wrong, I thought that 1/6+1/6 was 1/12. Thanks a lot for checking my work.
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    Double Integrals

    Whoops, I forgot to add in a step. However, when I integrate that I get what I have shown in my question and when I evaluate it on the interval from 0 to 6 I am getting the wrong answer which makes me think that maybe I am doing something else wrong.
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    Double Integrals

    Homework Statement Evaluate the double integral ∫∫D xy dA where D is the triangular region with vertices (0,0) (6,0) (0,1). Homework Equations The Attempt at a Solution 0 <= x <= -\frac{1}{6}x+1 0 <= x <= 6 the first integral would be the integral from 0 to -1/6x+1 of xy with...
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    Curvature Question

    I don't think that would be it since the cross product formula says that for the middle number it is subtraction. Besides, it is being squared, even if it is negative the answer which results will be positive.
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    Curvature Question

    Homework Statement Find the curvature K(t) of the curve r(t) = (-4sin(t)) i + (-4sin(t)) j + (5cos(t)) k. Homework Equations K(t) = |r'(t) x r"(t)| / |r'(t)|3 The Attempt at a Solution r'(t) = (-4cos(t))i + (-4cos(t))j + (-5sin(t))k r"(t) = (4sin(t))i + 4sin(t))j + (-5cos(t))k |r'(t)| =...
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    Implicit Equatio for a Plane

    Homework Statement An implicit equation for the plane passing through the point (-2,5,-5) that is perpendicular to the line L(t) = <5+2t,3,4> is ...? Homework Equations a(x-x0) + b(y-y0) + c(z-z0) = 0 The Attempt at a Solution So in order to find the equation of the plane I would...
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    Equation of the Plane

    Homework Statement Find an equation of a plane containing the three points (1, -1, 0), (5, 4, 1), (5, 5, 3) in which the coefficient of x is 9. What does it mean when it says the coefficient of x is 9? Homework Equations a(x-x0) + b(y-y0) + c(z-z0) = 0 The Attempt at a Solution...
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    Cross Product and Normal Vector Related Question

    In response to LCKurtz, sorry but I don't see how it works conceptually. What it looks like to me is that you are getting the magnitude of the cross product on the direction vector of the line and the line from the line to Q. From what I understand about cross products is that this will give us...
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    Cross Product and Normal Vector Related Question

    Homework Statement Find the distance from the point (2, 4, 4) to the line x = 0, y = 4 + 3t, z = 4 + 2t. Homework Equations The cross product and the dot product and d = |n * b|/|n| The Attempt at a Solution So the distance from the point to the line is the line directly...
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    Uniform Sound and Its Power

    Ah, then the whole problem makes sense since m2 cancels out nicely at the end leaving me with just Watts. Thanks a lot!
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    Uniform Sound and Its Power

    Homework Statement A spherical source radiates sound uniformly in all directions. At a distance of 9 m, the sound intensity level is 100 dB. What power is radiated by this source? Just a simple answer check to see if my answer is reasonable. Homework Equations \beta = 10 db log (I/I0)...
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    Fluid Buoyancy Check

    Homework Statement A raft is made of 11 logs lashed together. Each is 35 cm in diameter and has a length of 5.6 m. How many people can the raft hold before they start getting their feet wet, assuming the average person has a mass of 70 kg? Do not neglect the weight of the logs. Assume the...
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    What is wrong with my Math

    Homework Statement So the original problem goes as the following. A uniform ladder of mass (m) and length (L) leans against a frictionless wall, see figure. If the coefficient of static friction between the ladder and the ground is 0.41, what is the minimum angle (q) between the ladder and...
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