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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

    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

    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

    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

    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

    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

    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

    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

    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

    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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    Max Height Homogenous Differential Equation

    Homework Statement Let y be the the solution of the initial value problem. The maximum value of y is? y" + y = -sin(2x) y(0) = 0 y'(0) = 0 Homework Equations N/A The Attempt at a Solution Factor out original equation to find general equation. r2 + 1 = 0 r = 0 + i; 0- i Thus...
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    Non Homogenous Differential Equation

    Homework Statement y"' - 9y" +18y' = 30ex y(0) = 16 y'(0) = 14 y"(0) = 11 Homework Equations n/a The Attempt at a Solution Factor Out r(r2 - 9r +18) r = 0; r = 6; r =3 General Equation y(x) = c0 + c1e3x + c2e6x y'(x) = 3c1e3x + 6c2e6x y"(x) = 9c1e3x + 36c2e6x c1 = (11 - 36c2)/9 y'(x) = 14 =...
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    Simple NonHomogenous Differential Equation Question

    Homework Statement y"' -9y" +18y" = 30ex. Does not matter what the initial conditions are for my question. Homework Equations N/A The Attempt at a Solution So I factored out the equation and got: r*(r2 - 9r + 18) Which gives roots r = 0, r = 3, r = 6. Which gives me the general...
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    Fourth Order Homogenous Differential Equation

    Homework Statement y4 - 6y3 + 9y2 y(0) = 19; y'(0) = 16; y"(0) = 9; y"'(0) = 0 Homework Equations N/A The Attempt at a Solution Factored out the equation and obtained the following roots. r2(r2-3) = 0 which gives r = 0 and r =3. Using those roots, I make the following general...
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