Below the sphere x 2y z2 = 1, using cylindrical or spherical coordinates, whichever seems more appropriate Recall that the centroid is the center of mass of the solid assuming constant density Solution In spherical coordinates, the regions are given by 0 ˚ ˇ=4;0 ˆ 1 Thus, we compute the volume in spherical coordinates Vol(E) = ZZZ EUse spherical coordinates to evaluate the triple integral e−x2−y2−z2 x2y2z2 dV, ∫∫∫ E where E is the region bounded by the spheres x2y2z2=25 and x2y2z2=81 Change to spherical coordinates ρ2=25 ρ2=81 (5 pts) Set up the Integral e−ρ2 ρ ρ2sinφdρdφdθ ρ=5 ∫9 φ=0 ∫π θ=0 ∫2π (5 pts) (5 pts) Evaluate the IntegralExample 5 Find the z coordinate of the center of mass of the solid consisting of the part of the hemisphere z = p 4 x 2 y2 inside the cylinder x y2 = 2x if the density ˆ = 1 Answer Again we try using cylindrical coordinates, this time from the start Note that x2 y2 = 2x is not centered at the origin x y = 2x z = 4 x y 2 2 2 2 x y z
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X^2+y^2+z^2=1 in spherical coordinates
X^2+y^2+z^2=1 in spherical coordinates-Spherical coordinates are somewhat more difficult to understand The small volume we want will be defined by Δρ, Δϕ , and Δθ, as pictured in figure 1561 The small volume is nearly box shaped, with 4 flat sides and two sides formed from bits of concentric spheres When Δρ, Δϕ , and Δθ are all very small, the volume of this little First, we need to recall just how spherical coordinates are defined The following sketch shows the relationship between the Cartesian and spherical coordinate systems Here are the conversion formulas for spherical coordinates x = ρsinφcosθ y = ρsinφsinθ z = ρcosφ x2y2z2 = ρ2 x = ρ sin
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The equation in cylindrical coordinates is Step 2 (b) Conversion from rectangular coordinates to spherical coordinates The equation is Substitute and in above equation The equation in cylindrical coordinates is Solution (a) The equation in spherical coordinates is (b) The equation in spherical coordinates isTranslating Spherical Coordinates to Cartesian Coordinates The next step is to develop a technique for transforming spherical coordinates into Cartesian coordinates, and viceversaWrite the equation in spherical coordinates (a) x2 y2 z2 = 16 (b) x2 − y2 − z2 = 1 Expert Answer 100% (26 ratings) Previous question Next question Get more help from Chegg Solve it with our calculus problem solver and calculator
2 (3 pts) Use spherical coordinates to evaluate the triple integral ∫∫∫E xex 2 y2 z2dV where E is the portion of the unit ball x2y2z2≤1 that lies in the first octant 3 (3 pts) Use the spherical coordinates to evaluate the volume of E where E is the solid that lies above the cone z =√x2y2 and below the sphere x2y2z2 =81Angular momentum in spherical coordinates Peter Haggstrom wwwgotohaggstromcom mathsatbondibeach@gmailcom 1 Introduction Angular momentum is a deep property and in courses on quantum mechanics a lot of time is devoted to commutator relationships and spherical harmonics However, manyUse spherical coordinates to find the volume below the sphere x2 y2 z2 = 1 and above the cone z = p x2 y2 Solution R = n (ρ,φ,θ) θ ∈ 0,2π, φ ∈ h 0, π 4 i, ρ ∈ 0,1 o The calculation is simple, the region is a simple section of a sphere V = Z 2π 0 Z π/4 0 Z 1 0 ρ2 sin(φ) dρ dφ dθ, V = hZ 2π 0 dθ ihZ π/4 0
This video explains how to set up a triple integral using spherical coordinates and then evaluate the triple integralhttp//mathispower4ucomUse cylindrical or spherical coordinates, whichever seems more appropriate 35 Find the volume and centroid of the solid E that lies above the cone z = x 2 y 2 and below the sphere x 2 y 2 z 2 = 1This video explains how to convert a rectangular equation (cone) to a spherical equationhttp//mathispower4ucom
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Spherical coordinates (r, θ, φ) as commonly used in physics (ISO convention) radial distance r (distance to origin), polar angle θ (angle with respect to polar axis), and azimuthal angle φ (angle of rotation from the initial meridian plane)The symbol ρ is often used instead of rThis spherical coordinates converter/calculator converts the rectangular (or cartesian) coordinates of a unit to its equivalent value in spherical coordinates, according to the formulas shown above Rectangular coordinates are depicted by 3 values, (X, Y, Z) When converted into spherical coordinates, the new values will be depicted as (r, θ, φ)Write the equation in spherical coordinates 9 (a) x 2 y 2 z 2 = 9 (b) x 2 – y 2 – z 2 = 1 Buy Find launch Multivariable Calculus 8th Edition James Stewart Publisher Cengage Learning ISBN Buy Find launch Multivariable Calculus The spherical coordinates of a point are (8, /4, Ch 15 Identify the
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Example 1586 Setting up a Triple Integral in Spherical Coordinates Set up an integral for the volume of the region bounded by the cone z = √3(x2 y2) and the hemisphere z = √4 − x2 − y2 (see the figure below) Figure 15 A region bounded below by a cone and above by a hemisphere Solution So, given a point in spherical coordinates the cylindrical coordinates of the point will be, r = ρsinφ θ = θ z = ρcosφ r = ρ sin φ θ = θ z = ρ cos φ Note as well from the Pythagorean theorem we also get, ρ2 = r2 z2 ρ 2 = r 2 z 2 Next, let's find the Cartesian coordinates of the same point The coordinate in the spherical coordinate system is the same as in the cylindrical coordinate system, so surfaces of the form are halfplanes, as before Last, consider surfaces of the form The points on these surfaces are at a fixed angle from the z
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Just as polar coordinates gave us a new way of describing curves in the plane, in this section we will see how cylindrical and spherical coordinates give us new ways of describing surfaces and regions in space † † margin Figure 1471 Illustrating the principles behind cylindrical coordinates Spherical coordinates and finding the boundaries for $\rho$ 1 Given two points on a unit sphere, how to express their angular difference in spherical coordinates?;z), where rand are polar coordinates of the projection of Ponto the xyplane and zis the directed distance from the xyplane to P Figure 1 A point expressed in cylindrical coordinates
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Cartesian Cylindrical Spherical Cylindrical Coordinates x = r cosθ r = √x2 y2 y = r sinθ tan θ = y/x z = z z = z x2 y2 = 25 to cylindrical coordinates b) x2 y2 z2 = 1 to spherical coordinates c) ρ = 2cos φ to cylindrical coordinates 8 EX 4 Make the required change in the given equation (continued) Spherical form r=cos phi csc^2 theta Cylindrical form r=z csc^2theta The conversion formulas, Cartesian to spherical (x, y, z)=r(sin phi cos theta, sin phi sin theta, cos phi), r=sqrt(x^2y^2z^2) Cartesian to cylindrical (x, y, z)=(rho cos theta, rho sin theta, z), rho=sqrt(x^2y^2) Substitutions in x^2y^2=z lead to the forms in the answer Note the nuancesExample 65 Find the spherical equation for the hyperboloid of two sheets x2¡y2¡z2=1 Solution By direct substitution, we obtain, under the standard spherical coordinate system (‰sin`cosµ)2¡(‰sin`sinµ)2¡(‰cos`)2=1 or ‰ 2 ¡ sin`cos2µ¡sin`sin2µ¡cos2` ¢ =1 Example 66 Find a rectangular equation for the surface whose spher
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