Use spherical coordinates to evaluate where B is the ball

Question

Use spherical coordinates to evaluate  where B is the ball

Quizplus · Accepted Answer

In spherical coordinates, we have:
x = ρsinφcosθ
y = ρsinφsinθ
z = ρcosφ

where ρ is the radius, φ is the polar angle (measured from the positive z-axis), and θ is the azimuthal angle (measured from the positive x-axis).

In this problem, we have a ball with radius 1 centered at the origin:
ρ ≤ 1

To evaluate the given expression, we need to find the triple integral over the ball B:
∫∫∫B 11eaa3e5_54cb_c536_9f8f_c1880fc8b380_TB5971_11 dV

Using spherical coordinates, we have:

∫∫∫B 11eaa3e5_54cb_c536_9f8f_c1880fc8b380_TB5971_11 dV = ∫φ=0φ=π/2 ∫θ=0θ=2π ∫ρ=0ρ=1 11eaa3e5_54cb_c536_9f8f_c1880fc8b380_TB5971_11 ρ^2sinφ dρ dθ dφ
= 11eaa3e5_54cb_c537_9f8f_177ae989d756_TB5971_11 ∫φ=0φ=π/2 ∫θ=0θ=2π sinφ dθ dφ
= 11eaa3e5_54cb_c538_9f8f_dbf9e6117534_TB5971_11 ∫φ=0φ=π/2 dφ
= 11eaa3e5_54cb_c539_9f8f_177ae989d756_TB5971_11 [φ]φ=0φ=π/2
= 11eaa3e5_54cb_c539_9f8f_177ae989d756_TB5971_11 (π/2)

Therefore, the answer is B: 64 11eaa3e5_54cb_c539_9f8f_177ae989d756_TB5971_11.

Use Spherical Coordinates to Evaluate Where B Is the Ball