Question

a child has made a cubical region R which is bounded by the coordinate planes and the planes x=4 y=4 and z=4 in the first octant only. determine the average value of the function f=x^3y^3z^3 over this region R

Answer 1

Average=$\frac{1}{A}\int\int\int_vf(x,y,z)dxdydz$ $Average=\frac{1}{64}\int_0^4\int_0^4\int_0^4(x^3y^3z^3)dxdydz\\ Average=\frac{1}{64} \int_0^4\int_0^4[\frac{x^4}{4}y^3z^3]_0^4dydz\\ Average=\frac{1}{64} \int_0^4\int_0^4[\frac{4^4}{4}y^3z^3]_0^4dydz\\ Average= \frac{1}{64}\int_0^4\int_0^4[64y^3z^3]dydz\\ Average=\frac{1}{64} \int_0^4\int_0^4[64\frac{y^4}{4}z^3]_0^4dz\\ Average= \frac{1}{64} \int_0^4[64\times 64z^3]_0^4dz\\ Average=\frac{1}{64} [64\times 64\frac{z^4}{4}]_0^4dz\\ Average=\frac{1}{64}[64\times 64\times64]\\ Average=4096$