Use a CAS integration utility to evaluate the triple integral of the given function over the specified solid region.
-$F ( x , y , z ) = x ^ { 2 } y ^ { 4 } z ^ { 2 }$ over the cylinder bounded by $x ^ { 2 } + y ^ { 2 } \leq 49$ and the planes $z = - 7 , z = 10$
A) $\frac { 7,742,127,743 } { 236 } \pi$
B) $\frac { 7,742,127,743 } { 192 } \pi$
C) $\frac { 7,742,127,743 } { 208 } \pi$
D) $\frac { 158,002,607 } { 4 } \pi$

Question

Quizplus · Accepted Answer

The triple integral of $F(x, y, z) = x^2y^4z^2$ over the specified region can be computed by converting to cylindrical coordinates due to the symmetry of the problem. The bounds for $r$ are from 0 to 7 (the radius of the cylinder), $\theta$ ranges from 0 to $2\pi$ (completing a full circle), and $z$ ranges from -7 to 10 (the height of the cylinder). The integral in cylindrical coordinates becomes $\int_{0}^{2\pi} \int_{0}^{7} \int_{-7}^{10} r^2 \cdot r^4 \cdot z^2 \cdot r \, dz \, dr \, d\theta$, where $r^2$ comes from the conversion of $x^2 + y^2$ and the extra $r$ is the Jacobian determinant for the change of variables. The result of this integral is $\frac { 7,742,127,743 } { 192 } \pi$, which matches choice B.

Use a CAS Integration Utility to Evaluate the Triple Integral F(x,y,z)=x2y4z2F ( x , y , z ) = x ^ { 2 } y ^ { 4 } z ^ { 2 }F(x,y,z)=x2y4z2

Use a CAS Integration Utility to Evaluate the Triple Integral $F ( x , y , z ) = x ^ { 2 } y ^ { 4 } z ^ { 2 }$