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Write the equation in spherical coordinates (a) x2y2 z2 = 16 x 2 y 2 z 2 = 16 (b) x2−y2 −z2 = 1 x 2 − y 2 − z 2 = 1 Using Spherical Coordinates Let P (x,y,z) be any point in the plane, thenThe spherical coordinates of a point \(M\left( {x,y,z} \right)\) are defined to be the three numbers \(\rho, \varphi, \theta,\) where It is easier to calculate triple integrals in spherical coordinates when the region of integration \(U\) is a ball (or some portion of it) and/or when the integrand is a kind of \(f\left( {{x^2} {y^2} {z
X^2+y^2+z^2=16 in spherical coordinates
X^2+y^2+z^2=16 in spherical coordinates-Use spherical coordinates to evaluate the triple integral \displaystyle \iiint_E \, \frac{e^{(x^{2} y^{2} z^{2})}}{\sqrt{x^{2} y^{2} z^{2}}} \, dV, where E2 We can describe a point, P, in three different ways Cartesian Cylindrical Spherical Cylindrical Coordinates x = r cosθ r = √x2 y2 y = r sinθ tan θ = y/x z = z z = z Spherical Coordinates x = ρsinφcosθ ρ = √x2 y2 z2 y = ρsinφsinθ tan θ = y/x z = ρcosφ cosφ = √x2 y2 z2 z
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Use spherical coordinates to integrate f(x,y,z) = \sqrt{x^2 y^2 z^2} over the solid above the cone z = \sqrt{3x^2 3y^2} and inside the sphere x^2 y^2 z^2 = 4 View Answer A pole isTranslating between coordinates To translate between coordinate systems, consider the right triangle depicted in Fig ure 3 We have r= ˆcos(ˇ=2˚) = ˆ(cos(ˇ=2)cos˚sin(ˇ=2)sin˚) = ˆ(01 sin˚) = ˆsin˚ Thus x= rcos = ˆsin˚cos We compute yand zsimilarly x= ˆsin˚cos ˆ2= x2y2z2 y= ˆsin˚sin tan˚= p x2y2Triple integral in cylindrical coordinates (Sect 156) Example Use cylindrical coordinates to find the volume of a curved wedge cut out from a cylinder (x −
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 =81The spherical coordinates of P = (x ,y z) in the first quadrant are ρ = p x2 y2 z2, θ = arctan y x , and φ = arctan p x2 y2 z Spherical coordinates in R3 Example Use spherical coordinates to express region between the sphere x2 y2 z2 = 1 and the cone z = p x2 y2Some equations in spherical coordinates Sphere x2 y2 z2 = a2 ⇒ ρ = a Cone z2 = a2(x2 y2) ⇒ cos2(φ) = a2 sin2(φ) Cylinder x2 y2 = a2 ⇒ r = a or ρsin(φ) =
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We use the formulas expressing Cartesian in terms of spherical coordinates (setting ρ = a since (x,y,z) is on the sphere) (10) x = asinφcosθ, y = asinφsinθ, z = acosφ We can now calculate the flux integral (3) By (8) and (9), the integrand is F·ndS = 1 a (x2z y2z z2z)·a2sinφdφdθAbove and below by the sphere x2 y2 z2 = 9 and inside the cylinder x2 y2 = 4 z y x Page 5 of 18 V 0 2 This is a candidate for Spherical Coordinates 0 2 2 φ 16 3 ()sin()
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