Find the volume of the solid outside of the cone z^2 = x^2 + y^2 and inside the cylinder x^2 + y^2 = 4.
Question:
Find the volume of the solid outside of the cone {eq}z^2 = x^2 + y^2 {/eq} and inside the cylinder {eq}x^2 + y^2 = 4 {/eq}.
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Answers (1)
RhodaApril 19, 2023 в 06:27
The given equations represent a cone and a cylinder with the same base radius of 2. The cone has a height of 2 and the cylinder has a height of 4, so the cone is completely inside the cylinder. To find the volume of the solid outside the cone and inside the cylinder, we first find the volume of the cylinder and subtract the volume of the cone from it.
The volume of the cylinder is given by V_cyl = ?r^2h, where r is the radius of the base and h is the height. Substitute r = 2 and h = 4 to get V_cyl = 16?.
To find the volume of the cone, we use the formula V_cone = (1/3)?r^2h, where r is the radius of the base and h is the height. Since the cone is completely inside the cylinder, its radius and height are both less than those of the cylinder. Therefore, we need to find the height of the cone by setting the equations for the cone and cylinder equal to each other:
z^2 = x^2 + y^2 => z^2 = 4 - y^2 => y^2 = 4 - z^2
Substitute this expression for y^2 into the equation for the cone:
z^2 = x^2 + y^2 => z^2 = x^2 + (4 - z^2) => x^2 = z^2 - 4
This gives us the equation of a parabola in the xz-plane. The vertex of the parabola is at (0,0) and it opens downward, so the maximum value of z is when x = 0, which is z = 2. Therefore, the height of the cone is 2 units.
Substitute r = 2 and h = 2 into the formula for the volume of a cone to get V_cone = (4/3)?.
Finally, the volume of the solid outside the cone and inside the cylinder is given by:
V = V_cyl - V_cone = 16? - (4/3)? = (44/3)?
Answer: The volume of the solid outside of the cone {eq}z^2 = x^2 + y^2 {/eq} and inside the cylinder {eq}x^2 + y^2 = 4 {/eq} is (44/3)?.
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