Evaluate the integral.
-$$\int _ { 0 } ^ { 4 } \int _ { 0 } ^ { 8 ( 1 - x / 4 ) } \int _ { 0 } ^ { 7 ( 1 - x / 4 - y / 8 ) } x y z d z d y d x$$
A) $\frac { 12544 } { 81 }$
B) $\frac { 3136 } { 15 }$
C) $\frac { 6272 } { 45 }$
D) $\frac { 3136 } { 45 }$

Question

Quizplus · Accepted Answer

To solve the given triple integral, we integrate with respect to $z$, then $y$, and finally $x$. The bounds for $z$ are from $0$ to $7(1 - x/4 - y/8)$, for $y$ are from $0$ to $8(1 - x/4)$, and for $x$ are from $0$ to $4$.First, integrate with respect to $z$:$$\int_{0}^{7(1 - x/4 - y/8)} xyz \, dz = \frac{1}{2}xy[7(1 - x/4 - y/8)]^2$$This simplifies to:$$\frac{49}{2}xy(1 - x/4 - y/8)^2$$Next, integrate with respect to $y$:$$\int_{0}^{8(1 - x/4)} \frac{49}{2}xy(1 - x/4 - y/8)^2 \, dy$$This step requires expanding the quadratic term and integrating term by term. The integration will yield a polynomial in $x$.Finally, integrate with respect to $x$:$$\int_{0}^{4} [\text{Polynomial in } x] \, dx$$This final step yields the value of the integral.The correct answer is obtained by performing these steps accurately, which results in $\frac{3136}{45}$, making option D the correct choice. This process involves algebraic manipulation and integration techniques common in calculus.

Evaluate the Integral ∫04∫08(1−x/4)∫07(1−x/4−y/8)xyzdzdydx\int _ { 0 } ^ { 4 } \int _ { 0 } ^ { 8 ( 1 - x / 4 ) } \int _ { 0 } ^ { 7 ( 1 - x / 4 - y / 8 ) } x y z d z d y d x∫04​∫08(1−x/4)​∫07(1−x/4−y/8)​xyzdzdydx

Evaluate the Integral $\int _ { 0 } ^ { 4 } \int _ { 0 } ^ { 8 ( 1 - x / 4 ) } \int _ { 0 } ^ { 7 ( 1 - x / 4 - y / 8 ) } x y z d z d y d x$