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The volume of the cylinder x2 + y2 = a2 bounded below by z = 0 and bounded above by z = h is given by
- a)πah
- b)πa2h
- c)1/3πa3h
- d)None of these
Correct answer is option 'B'. Can you explain this answer?
Most Upvoted Answer
The volume of the cylinder x2 + y2 = a2 bounded below by z = 0 and bou...
The equation of the cylinder is
x2 + y2 = a2
The equation of surface CDE is z = h
So, the required volume is
Let x = a sin θ
implies dx = a cosθ dθ
So, volume V
x2 + y2 = a2
The equation of surface CDE is z = h
So, the required volume is
Let x = a sin θ
implies dx = a cosθ dθ
So, volume V
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The volume of the cylinder x2 + y2 = a2 bounded below by z = 0 and bou...
To find the volume of the cylinder, we can use triple integration.
Since the cylinder is described by the equation x^2 + y^2 = a^2, we can express the bounds for x and y in terms of polar coordinates:
0 ≤ r ≤ a
0 ≤ θ ≤ 2π
For the z-coordinate, the cylinder is bounded below by z = 0 and above by z = h, so the bounds for z are:
0 ≤ z ≤ h
The volume can be calculated using the triple integral:
V = ∫∫∫ dV
Where dV is the infinitesimal volume element. In cylindrical coordinates, dV can be expressed as:
dV = r dz dr dθ
Substituting the bounds for r, θ, and z, the volume becomes:
V = ∫[0 to 2π]∫[0 to a]∫[0 to h] r dz dr dθ
Evaluating this integral will give the volume of the cylinder.
Since the cylinder is described by the equation x^2 + y^2 = a^2, we can express the bounds for x and y in terms of polar coordinates:
0 ≤ r ≤ a
0 ≤ θ ≤ 2π
For the z-coordinate, the cylinder is bounded below by z = 0 and above by z = h, so the bounds for z are:
0 ≤ z ≤ h
The volume can be calculated using the triple integral:
V = ∫∫∫ dV
Where dV is the infinitesimal volume element. In cylindrical coordinates, dV can be expressed as:
dV = r dz dr dθ
Substituting the bounds for r, θ, and z, the volume becomes:
V = ∫[0 to 2π]∫[0 to a]∫[0 to h] r dz dr dθ
Evaluating this integral will give the volume of the cylinder.
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