The quantity of the strong is the indispensable under \$latex z= f( x, y)\$, with the wall surfaces provided by the aircrafts \$latex y= 0\$, \$latex z= 0\$ and also \$latex x= 3\$.

The primary step is to figure out the crossways of \$latex f( x, y)= 4-y ^ 2\$ with the coordinate axes, in order to locate the limitations of assimilation.

Junction of \$latex f( x, y)\$ with z-axis

Taking \$latex y = 0\$, we have: \$latex z =4-0 ^ 2= 4\$

Therefore, the junction of the allegorical cyndrical tube with the z-axis is the factor \$latex (0,0,4)\$.

Junction of \$latex f( x, y)\$ with y-axis

Taking \$latex z= 0\$, we have:

\$ latex 4-y ^ 2= 0Rightarrow y =2\$

The favorable origin is taken considering that the junction with the favorable y-axis is looked for, which is the factor \$latex (0,2,0)\$.

The aircraft \$latex y= 0\$ restricts the strong, considering that it remains in the very first octant, according to the declaration, consequently, the base of the strong is a rectangular shape with measurements \$latex x= 3\$ and also \$latex y = 2\$, while both continuing to be wall surfaces are provided by the aircrafts \$latex x= 0\$ and also \$latex x= 3\$.

The resulting strong is received the number listed below:

The quantity of the strong represents the indispensable:

\$\$ V= int_a ^ bint_c ^ d f( x, y) dydx\$\$

Where \$latex y\$ ranges \$latex 0\$ and also \$latex 2\$, while \$latex x\$ ranges \$latex 0\$ and also \$latex 3\$, consequently:

\$\$ V= int_0 ^ 3left[int_0^2 (4 -y^2)dyright]dx\$\$

First, the internal indispensable is determined:

\$\$ int_0 ^ 2 (4 -y ^ 2) dy= int_0 ^ 2 4dy-int_0 ^ 2 y ^ 2dy\$\$

\$\$ 4yBig|_ 0 ^ 2-dfrac {y ^ 3} {3} Huge|_ 0 ^ 2\$\$

\$\$= 8-dfrac {8} {3} =dfrac {16} {3} \$\$

The outcome is replaced in the quantity indispensable brace:

\$\$ V= int_0 ^ 3left( dfrac {16} {3} right) dx\$\$

\$\$= dfrac {16} {3} int_0 ^ 3dx= left( dfrac {16} {3} right) xBig|_ 0 ^ 3= 16\$\$