Find the volume under the paraboloid $z = x ^ { 2 } + y ^ { 2 }$ above the region bounded by the x-axis, the y-axis, and the line x + y = 1.
A) $\frac { 1 } { 6 }$

B) $\frac { 1 } { 4 }$
C) $\frac { 1 } { 3 }$
D) $\frac { 1 } { 2 }$
E) $\frac { 2 } { 3 }$
F) $\frac { 3 } { 4 }$

G) $\frac { 5 } { 6 }$

H)1

Question

Find the volume under the paraboloid $z = x ^ { 2 } + y ^ { 2 }$ above the region bounded by the x-axis, the y-axis, and the line x + y = 1.
A) $\frac { 1 } { 6 }$

B) $\frac { 1 } { 4 }$ 
C) $\frac { 1 } { 3 }$ 
D) $\frac { 1 } { 2 }$ 
E) $\frac { 2 } { 3 }$ 
F) $\frac { 3 } { 4 }$

G) $\frac { 5 } { 6 }$

H)1

Quizplus · Accepted Answer

The volume is found by integrating $z = x^2 + y^2$ over the triangular region with vertices (0,0), (1,0), and (0,1). The double integral evaluates to $\frac{1}{6}$.

Find the Volume Under the Paraboloid z=x2+y2z = x ^ { 2 } + y ^ { 2 }z=x2+y2

Find the Volume Under the Paraboloid $z = x ^ { 2 } + y ^ { 2 }$