Find the volume of the solid bounded by the paraboloid $z = x ^ { 2 } + y ^ { 2 }$ and the plane $z = 4$ .
A)4$\pi$
B)8$\pi$
C)12$\pi$
D)16$\pi$
E)24$\pi$
F)32$\pi$
G)48$\pi$
H)64$\pi$

Question

Quizplus · Accepted Answer

The volume of the solid can be found by integrating the function $z = x^2 + y^2$ within the bounds set by $z = 4$. The intersection of $z = x^2 + y^2$ and $z = 4$ gives the boundary of the solid in the xy-plane, which is a circle of radius 2 (since $4 = x^2 + y^2$ implies $r^2 = 4$, where $r$ is the radius). The volume can be calculated using cylindrical coordinates or by evaluating the integral directly, leading to the formula for the volume of a cylinder, $\pi r^2 h$, where $r = 2$ and $h = 4$, giving a volume of $16\pi$.

Find the Volume of the Solid Bounded by the Paraboloid z=x2+y2z = x ^ { 2 } + y ^ { 2 }z=x2+y2

Find the Volume of the Solid Bounded by the Paraboloid $z = x ^ { 2 } + y ^ { 2 }$