The volume of the solid bounded $x = 0 , x = 1 - y ^ { 2 } , x ^ { 2 } + y ^ { 2 } = z$ and z = 0 is

A) $\frac { 3 } { 7 }$
B) $\frac { 4 } { 7 }$
C) $\frac { 5 } { 7 }$
D) $\frac { 6 } { 7 }$
E) $\frac { 8 } { 7 }$

Question

The volume of the solid bounded $x = 0 , x = 1 - y ^ { 2 } , x ^ { 2 } + y ^ { 2 } = z$ and z = 0 is

A) $\frac { 3 } { 7 }$
B) $\frac { 4 } { 7 }$
C) $\frac { 5 } { 7 }$
D) $\frac { 6 } { 7 }$
E) $\frac { 8 } { 7 }$

Quizplus · Accepted Answer

The volume of the solid can be found using triple integration. The bounds for $x$ are given as $0$ to $1 - y^2$, for $y$ as the intersection points of $x = 1 - y^2$ and $x^2 + y^2 = z$ projected on the $y$-axis, and for $z$ from $0$ to $x^2 + y^2$. The correct integration setup and calculation yield a volume of $\frac{4}{7}$, making choice B the correct answer.

The Volume of the Solid Bounded x=0,x=1−y2,x2+y2=zx = 0 , x = 1 - y ^ { 2 } , x ^ { 2 } + y ^ { 2 } = zx=0,x=1−y2,x2+y2=z

The Volume of the Solid Bounded $x = 0 , x = 1 - y ^ { 2 } , x ^ { 2 } + y ^ { 2 } = z$