1. Multiple integrals
(a) Calculate the integral of f(x,y,z) = xz^(2)e^(xyz) over the region
G = {(x,y,z) : x in [0,1], y in [0,2], z in [0,3]}.
Hint: one specific order of integration will be easier than the others, examine f and carefully consider which integral to do first and then second before starting.
(b) Evaluate the integral of f(x,y,z) = xyz over the region
RLG = {(r,θ,z) : 0 <= θ <= 2π, 0 <= z <= L, 0 <= r <= R(z)/(L)}. zz_(c) = (M_(z))/(M) M = ∭_(G)dV M_(epsi) = ∭_(G)zdV ρ(x,y) = 10 - 3x - 5y + 4xy.
2. Multiple integrals
(a) Calculate the integral of fyzeyzovcr over the region G{ye0.1ye0.2ze{03} Hint: one specific order of integration uill be easier than the others, examine f and carefully consider which integral to do first and then second before starting. (b) Evaluate the integral of fryzryz over the region G{UrOVOyI}
(c) Consider a solid cone with base radius R and length L. In cylindrical coordinates, the cone may be described as occupying the region
G{T0OO2TOZLOTR/L}
Use triple integrals to find the coordinate of the cone's centroid. That is, find ze - M/M
(d) Find the center of mass of a rectangular sheet of material occupying
R{020K1}
and having a non-uniform density described by
10 - 3y + 4y