Unformatted text preview: PRINT NAME: Solutions Calculus IV [2443–002] Quiz III Tuesday, April 4, 2000 Q1]... Write the following triple integral out as a spherical coordinates triple integral. Z 3- 3 Z √ 9- x 2 Z √ 9- x 2- y 2 z ( x 2 + y 2 + z 2 ) dzdydx Soln: The region is precisely one quarter of a solid ball which is centered on the origin and has radius 3. The quarter is is above the xy-plane and to the positive y half of the xz-plane. The spherical coordinates description of this region is just ≤ ρ ≤ 3 , ≤ θ ≤ π , ≤ φ ≤ π/ 2 . Noting that the integrand converts into ρ cos φ ( ρ 2 ), and remembering that dv = ρ 2 sin φdρdθdφ , we obtain Z π/ 2 Z π Z 3 ρ 5 cos φ sin φdρdθdφ. Q2]... Sketch the region which is described in the following triple integral. Z π/ 4 Z π/ 2 Z sec φ ρ 2 sin φdρdθdφ Soln: We build the region up from small blocks which radiate outwards from the origin ( ρ = 0) until the horizontal plane z = 1 ( ρ = sec φ ). We see this last fact from the definition of spherical coordinates as follows:)....
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- Winter '08
- Calculus, ... ..., Spherical coordinate system, Multiple integral