The sub-boxes will have dimensions [tex] \frac{2-0}{2} \times \frac{2-0}{2} \times \frac{2-0}{2} =1\times1\times1=1 \ cubic \ units[/tex]
x sub-intervals are 0 to 1 and 1 to 2. Midpoints are at [tex]x= \frac{1}{2} [/tex] and [tex]x= \frac{3}{4} [/tex]
y sub-intervals are 0 to 1 and 1 to 2. Midpoints are at [tex]y= \frac{1}{2} [/tex] and [tex]y= \frac{3}{4} [/tex]
z sub-intervals are 0 to 1 and 1 to 2. Midpoints are at [tex]z= \frac{1}{2} [/tex] and [tex]z= \frac{3}{4} [/tex]
Let [tex]f(x,y,z)=\cos{(xyz)}[/tex]
[tex] \int\limits \int\limits \int\limits {f(x,y,z)} \, dV \approx f\left( \frac{1}{2} , \frac{1}{2} , \frac{1}{2} \right)+f\left( \frac{1}{2} , \frac{1}{2} , \frac{3}{4} \right)+f\left( \frac{1}{2} , \frac{3}{4} , \frac{1}{2} \right)+f\left( \frac{1}{2} , \frac{3}{4} , \frac{3}{4} \right)[/tex]
[tex]+f\left( \frac{3}{4} , \frac{1}{2} , \frac{1}{2} \right)+f\left( \frac{3}{4} , \frac{1}{2} , \frac{3}{4} \right)+f\left( \frac{3}{4} , \frac{3}{4} , \frac{1}{2} \right)+f\left( \frac{3}{4} , \frac{3}{4} , \frac{3}{4} \right) \\ \\ \approx\cos{ \frac{1}{8} }+\cos{ \frac{3}{16} }+\cos{ \frac{3}{16} }+\cos{ \frac{9}{32} }+\cos{ \frac{3}{16} }+\cos{ \frac{9}{32} }+\cos{ \frac{9}{32} }+\cos{ \frac{27}{64} } \\ \\ \approx0.9922+0.9825+0.9825+0.9607+0.9825+0.9607+0.9607 \\ +0.9123 \\ \\ \approx\bold{7.734}[/tex]
