Copyright © Cengage Learning. All rights reserved. 15 Multiple IntegralsCopyright © Cengage Learning. All rights reserved. 15.10 Change of Variables in Triple Integrals3 3 Change of Variables in Triple Integrals Learning objective: § simplify triple integrals by a change of variables (and then evaluate)4 4 Triple Integrals We have seen a formula for a change of variables for double integrals. There is a similar change of variables formula for triple integrals. Let T be a transformation that maps a region S in uvw-space onto a region R in xyz-space by means of the equations x = g(u, v, w) y = h(u, v, w) z = k(u, v, w)5 5 Triple Integrals The Jacobian of T is the following 3 × 3 determinant: Under hypotheses similar to those in Theorem 9, we have the following formula for triple integrals:6 6 Example Use Formula 13 to derive the formula for triple integration in spherical coordinates. Solution: Here the change of variables is given by x = ρ sin φ cos θ y = ρ sin φ sin θ z = ρ cos φ We compute the Jacobian as follows:7 7 Example – Solution = cos φ (–ρ2 sin φ cos φ sin2 θ – ρ2 sin φ cos φ cos2 θ) – ρ sin φ (ρ sin2 φ cos2 θ + ρ sin2 φ sin2 θ) = –ρ2 sin φ cos2 φ – ρ2 sin φ sin2 φ = –ρ2 sin φ Since 0 ≤ φ ≤ π, we have sin φ ≥ 0. continued8 8 Example – Solution Therefore and Formula 13 gives f (x, y, z) dV = f (ρ sin φ cos θ, ρ sin φ sin θ, ρ cos φ) ρ2 sin φ dρ dθ dφ
View Full Document
Unlocking...