Please help me evaluate this integral, I know it needs to go into cylendrical coordinates but I cannot figure out what theta should be?
Transcribed Image Text:2 V16-х 4
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3/2
With differentiation, one of the major concepts of calculus. Integration involves the calculation of an integral, which is useful to find many quantities such as areas, volumes, and displacement.
Expert Solution
Step 1
We are given the integral,
We will be solving the above integral by converting it into cylindrical coordinates.
Note:
For the cylindrical co-ordinate, x = r cos(θ), y = r sin(θ), and z = z.
And dV = dz dy dx = r dz dr dθ.
Step 2
Here, 0 ≤ x ≤ 2,
0 ≤ y ≤ √(16 - x2),
and 0 ≤ z ≤ 4.
Notice that, 0 ≤ y ≤ √(16 - x2) => 0 ≤ y2 ≤ 16 - x2 (squaring both sides)
=> 0 ≤ x2 + y2 ≤ 16
=> 0 ≤ r2 ≤ 16
=> 0 ≤ r≤ 4
Further, 0 ≤ x ≤ 2 => 0 ≤ x ≤ 2
=> 0 ≤ r cos(θ) ≤ 2
=> 0 ≤ cos(θ) ≤ 1/2 (As 0 ≤ r≤ 4)
We know, cos-1(0) = π/2 and cos-1(1/2) = π/3.
=> π/3 ≤ θ ≤ π/2.
Summary:
Take, x = r cos(θ), y = r sin(θ), and z = z.
Then, dz dy dx = r dz dr dθ.
And here, 0 ≤ z ≤ 4, 0 ≤ r≤ 4, and π/3 ≤ θ ≤ π/2.
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