The region W is shown below.Write the limits of integration for $\int _ { w } f ( x , y , z ) d V$ in spherical coordinates.
A) $\int _ { W } f ( x , y , z ) d V = \int _ { 0 } ^ { \pi } \int _ { \pi / 2 } ^ { \pi } \int _ { 0 } ^ { 1 } f ( \rho \sin \phi \cos \theta , \rho \sin \phi \sin \theta , \rho \cos \phi ) \rho ^ { 2 } \sin \phi d \rho d \phi d \theta$
B) $\int _ { W } f ( x , y , z ) d V = \int _ { - 1 } ^ { 1 } \int _ { 0 } ^ { \sqrt { 1 - x ^ { 2 } } } \int _ { - \sqrt { 1 - x ^ { 2 } - y ^ { 2 } } } ^ { 0 } f ( x , y , z ) d z d y d x$
C) $\int _ { W } f ( x , y , z ) d V = \int _ { 0 } ^ { \pi } \int _ { 0 } ^ { 1 } \int _ { - \sqrt { 1 - r ^ { 2 } } } ^ { 0 } f ( r \cos \theta , r \sin \theta , z ) r d z d r d \theta$

Question

The region W is shown below.Write the limits of integration for $\int _ { w } f ( x , y , z ) d V$ in spherical coordinates. 
A) $\int _ { W } f ( x , y , z ) d V = \int _ { 0 } ^ { \pi } \int _ { \pi / 2 } ^ { \pi } \int _ { 0 } ^ { 1 } f ( \rho \sin \phi \cos \theta , \rho \sin \phi \sin \theta , \rho \cos \phi ) \rho ^ { 2 } \sin \phi d \rho d \phi d \theta$ 
B) $\int _ { W } f ( x , y , z ) d V = \int _ { - 1 } ^ { 1 } \int _ { 0 } ^ { \sqrt { 1 - x ^ { 2 } } } \int _ { - \sqrt { 1 - x ^ { 2 } - y ^ { 2 } } } ^ { 0 } f ( x , y , z ) d z d y d x$ 
C) $\int _ { W } f ( x , y , z ) d V = \int _ { 0 } ^ { \pi } \int _ { 0 } ^ { 1 } \int _ { - \sqrt { 1 - r ^ { 2 } } } ^ { 0 } f ( r \cos \theta , r \sin \theta , z ) r d z d r d \theta$

Quizplus · Accepted Answer

The Answer of The region W is shown below.Write the limits of integration for $\int _ { w } f ( x , y , z ) d V$ in spherical coordinates. 
A) $\int _ { W } f ( x , y , z ) d V = \int _ { 0 } ^ { \pi } \int _ { \pi / 2 } ^ { \pi } \int _ { 0 } ^ { 1 } f ( \rho \sin \phi \cos \theta , \rho \sin \phi \sin \theta , \rho \cos \phi ) \rho ^ { 2 } \sin \phi d \rho d \phi d \theta$ 
B) $\int _ { W } f ( x , y , z ) d V = \int _ { - 1 } ^ { 1 } \int _ { 0 } ^ { \sqrt { 1 - x ^ { 2 } } } \int _ { - \sqrt { 1 - x ^ { 2 } - y ^ { 2 } } } ^ { 0 } f ( x , y , z ) d z d y d x$ 
C) $\int _ { W } f ( x , y , z ) d V = \int _ { 0 } ^ { \pi } \int _ { 0 } ^ { 1 } \int _ { - \sqrt { 1 - r ^ { 2 } } } ^ { 0 } f ( r \cos \theta , r \sin \theta , z ) r d z d r d \theta$

The Region W Is Shown Below ∫wf(x,y,z)dV\int _ { w } f ( x , y , z ) d V∫w​f(x,y,z)dV

The Region W Is Shown Below $\int _ { w } f ( x , y , z ) d V$