Evaluate the triple integral $\iiint _ { E } x d V$ , where $E = \{ ( x , y , z ) \mid 0 \leq x \leq y , 0 \leq y \leq 1,0 \leq z \leq 1 \}$ .
A) $\frac { 1 } { 6 }$

B) $\frac { 1 } { 4 }$
C) $\frac { 1 } { 3 }$
D) $\frac { 1 } { 2 }$
E) $\frac { 2 } { 3 }$
F) $\frac { 1 } { 12 }$

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

H)1

Question

Evaluate the triple integral $\iiint _ { E } x d V$ , where $E = \{ ( x , y , z ) \mid 0 \leq x \leq y , 0 \leq y \leq 1,0 \leq z \leq 1 \}$ .
A) $\frac { 1 } { 6 }$

B) $\frac { 1 } { 4 }$ 
C) $\frac { 1 } { 3 }$ 
D) $\frac { 1 } { 2 }$ 
E) $\frac { 2 } { 3 }$
F) $\frac { 1 } { 12 }$

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

H)1

Quizplus · Accepted Answer

The integral can be set up as $\int_{0}^{1} \int_{0}^{y} \int_{0}^{1} x \, dz \, dx \, dy$. Integrating with respect to $z$ first gives $x$, since $z$ ranges from 0 to 1 and does not affect $x$. Then, integrating $x$ from 0 to $y$ gives $\frac{1}{2}y^2$. Finally, integrating $\frac{1}{2}y^2$ from 0 to 1 gives $\frac{1}{6}$, which is the volume under the curve within the specified bounds.

Evaluate the Triple Integral ∭ExdV\iiint _ { E } x d V∭E​xdV

Evaluate the Triple Integral $\iiint _ { E } x d V$