Set up (but do not evaluate)an iterated integral to compute the mass of the solid paraboloid bounded by $z = x ^ { 2 } + y ^ { 2 }$ and z = 1, if the density is given by $\delta$(x, y, z)= z². A) $\text { Mass } = \int _ { - 1 } ^ { 1 } \int _ { - \sqrt { 1 - x ^ { 2 } } } ^ { \sqrt { 1 - x ^ { 2 } } } \int _ { x ^ { 2 } + y ^ { 2 } } ^ { 1 } z ^ { 2 } d z d x d y$ B) $\text { Mass } = \int _ { 0 } ^ { 1 } \int _ { 0 } ^ { \sqrt { 1 - x ^ { 2 } } } \int _ { x ^ { 2 } + y ^ { 2 } } ^ { 1 } z ^ { 2 } d z d y d x$ C) $\operatorname { Mass } = \int _ { - 1 } ^ { 1 } \int _ { - \sqrt { 1 - x ^ { 2 } } } ^ { \sqrt { 1 - x ^ { 2 } } } \int _ { 0 } ^ { 1 } z ^ { 2 } d z d y d x$ D) $\text { Mass } = \int _ { - 1 } ^ { 1 } \int _ { - \sqrt { 1 - x ^ { 2 } } } ^ { \sqrt { 1 - x ^ { 2 } } } \int _ { x ^ { 2 } + y ^ { 2 } } ^ { 1 } z ^ { 2 } d z d y d x$ E) $\text { Mass } = \int _ { 0 } ^ { 1 } \int _ { - \sqrt { 1 - x ^ { 2 } } } ^ { \sqrt { 1 - x ^ { 2 } } } \int _ { x ^ { 2 } + y ^ { 2 } } ^ { 1 } z ^ { 2 } d z d y d x$

Question

Quizplus · Accepted Answer

The correct setup for the iterated integral involves integrating over the circular base in the xy-plane, which is defined by $x^2 + y^2 \leq 1$, and then integrating in the z-direction from the paraboloid $z = x^2 + y^2$ up to the plane $z = 1$. The density function is given as $z^2$, so it needs to be integrated with respect to z first, then y, and finally x. The limits for x are from -1 to 1, and for y, they are from $-\sqrt{1 - x^2}$ to $\sqrt{1 - x^2}$, to cover the entire circular base. This is correctly represented in option D.

Set Up (But Do Not Evaluate)an Iterated Integral to Compute z=x2+y2z = x ^ { 2 } + y ^ { 2 }z=x2+y2

Set Up (But Do Not Evaluate)an Iterated Integral to Compute $z = x ^ { 2 } + y ^ { 2 }$