Consider the half of the sphere of radius 2 centered at the origin which lies above the xy-plane (a hemisphere). Suppose the density at a point is precisely the distance from the origin. Find the total mass up to three decimal places. (Hint: use spherical coordinates.)

25.133 units

Step-by-step explanation:

Since the density ρ = r, our mass is

m =  ∫∫∫r³sinθdΦdrdθ. We integrate from θ = 0 to π (since it is a hemisphere), Φ = 0 to 2π and r = 0 to 2  and  the maximum values of r = 2 in those directions. So

m =∫∫[∫r³sinθdΦ]drdθ      

m = ∫[∫2πr³sinθdθ]dr     ∫dФ = 2π

m = ∫2πr³∫sinθdθ]dr

m = 2π∫r³dr    ∫sinθdθ = 1

m = 2π × 4     ∫r³dr = 4

m = 8π units

m = 25.133 units

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step-by-step explanation:

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