16.4 Ex 48 - 51

16.4 Ex 48 - 51 - S E C T I O N 16.4 Integration in Polar...

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Unformatted text preview: S E C T I O N 16.4 Integration in Polar, Cylindrical, and Spherical Coordinates (ET Section 15.4) 969 48. Calculate the volume of the sphere x 2 + y 2 + z 2 = a 2 , using both spherical and cylindrical coordinates. SOLUTION Spherical coordinates: In the entire sphere of radius a , we have W : ≤ θ ≤ 2 π , ≤ φ ≤ π , ≤ ρ ≤ a Using triple integral in spherical coordinates we get V = ZZZ W 1 dV = Z 2 π Z π Z a ρ 2 sin φ d ρ d φ d θ = Z a ρ 2 d ρ ¶ Z π sin φ d φ ¶ Ã Z 2 π 1 d θ ! = Ã ρ 3 3 ¯ ¯ ¯ ¯ a ! − cos φ ¯ ¯ ¯ ¯ π ¶ Ã θ ¯ ¯ ¯ ¯ 2 π ! = a 3 3 · 2 · 2 π = 4 π a 3 3 Cylindrical coordinates: The projection of W onto the xy-plane is the circle of radius a , that is, D : ≤ θ ≤ 2 π , ≤ r ≤ a The upper surface is z = p a 2 − ( x 2 + y 2 ) = p a 2 − r 2 and the lower surface is z = − p a 2 − r 2 . Therefore, W has the following description in cylindrical coordinates: W : ≤ θ ≤ 2 π , ≤ r ≤ a , − p a 2 − r 2 ≤ z ≤ p a 2 − r 2 We obtain the following integral: V = Z 2 π Z a Z √ a 2 − r 2 − √ a 2 − r 2 r dz dr d θ = Z 2 π Z a rz ¯ ¯ ¯ ¯ √ a 2 − r 2 z =− √ a 2 − r 2 dr d θ = Z 2 π Z a 2 r p a 2 − r 2 dr d θ (1) We compute the inner integral using the substitution...
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16.4 Ex 48 - 51 - S E C T I O N 16.4 Integration in Polar...

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