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The volume of an object expressed in spherical coordinates is given by 
sin φ dr dφ dθ. The value of the integral is
sin φ dr dφ dθ. The value of the integral is- a)π/3
- b)2π/3
- c)π/6
- d)π/4
Correct answer is option 'A'. Can you explain this answer?
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The volume integral in spherical coordinates is given by:
V = ∫02π ∫0π/3 ∫01 r² sin(φ) dr dφ dθ.
Step 1: Integrate with respect to r
The inner integral is with respect to r from 0 to 1:
∫01 r² dr = [ r³/3 ]01 = 1/3.
Substituting this result, we now have:
V = ∫02π ∫0π/3 (1/3) sin(φ) dφ dθ.
Step 2: Integrate with respect to φ
Now we integrate with respect to φ from 0 to π/3:
∫0π/3 sin(φ) dφ = [ -cos(φ) ]0π/3 = -cos(π/3) + cos(0).
Since cos(π/3) = 1/2 and cos(0) = 1, this becomes:
= -1/2 + 1 = 1/2.
Substituting this result, we get:
V = ∫02π (1/3) * (1/2) dθ = (1/6) ∫02π dθ.
Step 3: Integrate with respect to θ
Now we integrate with respect to θ from 0 to 2π:
∫02π dθ = 2π.
So, we have:
V = (1/6) * 2π = π/3.
Conclusion:
The value of the integral is:
A: π/3.
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The volume of an object expressed in spherical coordinates is given bysin φ dr dφ dθ.The value of the integral isa)π/3b)2π/3c)π/6d)π/4Correct answer is option 'A'. Can you explain this answer?
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