Solve the problem.
-the cylinder $x ^ { 2 } + y ^ { 2 } = 64$ and the plane $z = 6$.
A) $\int _ { 0 } ^ { 8 } \int _ { 0 } ^ { \sqrt { 64 - x ^ { 2 } } } \int _ { 0 } ^ { 6 } d z d y d x$
B) $\int _ { 0 } ^ { 8 } \int _ { 0 } ^ { \sqrt { 64 - y ^ { 2 } } } \int _ { 0 } ^ { 6 - y } d z d y d x$
C) $\int _ { 0 } ^ { 8 } \int _ { 0 } ^ { \sqrt { 64 - x ^ { 2 } } } \int _ { 0 } ^ { y } d z d y d x$
D) $\int _ { 0 } ^ { 8 } \int _ { 0 } ^ { \sqrt { 64 - y ^ { 2 } } } \int _ { 0 } ^ { 6 } d z d y d x$

Question

Quizplus · Accepted Answer

The correct integral setup for the volume of the cylinder $x^2 + y^2 = 64$ under the plane $z = 6$ involves integrating $z$ from 0 to 6, $y$ from 0 to $\sqrt{64 - x^2}$, and $x$ from 0 to 8, considering only a quarter of the cylinder due to the limits of integration and symmetry.

Solve the Problem x2+y2=64x ^ { 2 } + y ^ { 2 } = 64x2+y2=64

Solve the Problem $x ^ { 2 } + y ^ { 2 } = 64$