Triple Integrals — Question 6

Question 6 – Solution

Strategy. Pair $x$ with $-x$; the nonconstant term is odd in $x$.

Step 1: Cancellation$x^3yz^2$ integrates to zero over the symmetric $x$-interval.

Step 2: Constant termThe box volume is $(4)(2)(3)=24$, so $\boxed{\iiint_B(x^3yz^2+4)dV=4(24)=96}$.

VerificationReflection $(x,y,z)\mapsto(-x,y,z)$ preserves $B$ and reverses only the nonconstant term. Without that regional symmetry the cancellation would not follow.