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The volume of the portion of the solid cylinder x2 + y2 ≤ 2 bounded above by the surface z = x2 + y2 and bounded below by the xy- plane is
- a)π
- b)2π
- c)3π
- d)4π
Correct answer is option 'B'. Can you explain this answer?
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The volume of the portion of the solid cylinder x2 + y2 2 bounded abov...
Solution:
Given, the equation of the surface is z = x^2 + y^2 and it is a solid cylinder.
To find the volume of the portion of the solid cylinder, we need to find the limits of integration.
Limits of integration for x and y:
The solid cylinder is bounded by the xy-plane, so the limits of integration for x and y are from -r to r, where r is the radius of the cylinder.
Limits of integration for z:
The surface z = x^2 + y^2 is the upper boundary of the solid cylinder. The lower boundary is the xy-plane, which is z = 0.
So, the limits of integration for z are from 0 to x^2 + y^2.
Volume of the portion of the solid cylinder:
The volume of the portion of the solid cylinder can be found by integrating the equation of the surface over the limits of integration.
V = ∫∫∫ dV
V = ∫∫∫ dzdydx
V = ∫-r^r ∫-r^r ∫0^(x^2+y^2) dzdydx
V = ∫-r^r ∫-r^r (x^2+y^2) dydx
V = ∫-r^r (x^2y + y^3/3)|-r^r dx
V = ∫-r^r [(2/3)r^3 + (2/3)r^3] dx
V = (4/3)r^3 ∫-r^r dx
V = (4/3)r^3 [x]|-r^r
V = (4/3)r^3 [r - (-r)]
V = (4/3)r^3 (2r)
V = (8/3)r^4
Therefore, the volume of the portion of the solid cylinder is (8/3)r^4, which is option B.
Given, the equation of the surface is z = x^2 + y^2 and it is a solid cylinder.
To find the volume of the portion of the solid cylinder, we need to find the limits of integration.
Limits of integration for x and y:
The solid cylinder is bounded by the xy-plane, so the limits of integration for x and y are from -r to r, where r is the radius of the cylinder.
Limits of integration for z:
The surface z = x^2 + y^2 is the upper boundary of the solid cylinder. The lower boundary is the xy-plane, which is z = 0.
So, the limits of integration for z are from 0 to x^2 + y^2.
Volume of the portion of the solid cylinder:
The volume of the portion of the solid cylinder can be found by integrating the equation of the surface over the limits of integration.
V = ∫∫∫ dV
V = ∫∫∫ dzdydx
V = ∫-r^r ∫-r^r ∫0^(x^2+y^2) dzdydx
V = ∫-r^r ∫-r^r (x^2+y^2) dydx
V = ∫-r^r (x^2y + y^3/3)|-r^r dx
V = ∫-r^r [(2/3)r^3 + (2/3)r^3] dx
V = (4/3)r^3 ∫-r^r dx
V = (4/3)r^3 [x]|-r^r
V = (4/3)r^3 [r - (-r)]
V = (4/3)r^3 (2r)
V = (8/3)r^4
Therefore, the volume of the portion of the solid cylinder is (8/3)r^4, which is option B.
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