Solving a triple integral using the change of variables

 In the previous posts we spent time establishing formulas to solve a triple integral using a change of variables. Let's now solve an example.

Example

Solution

Let's apply the formula:

Let's take the following steps to apply the formula:

1) Let's find x, y, z from the given values of u, v, w. This leads to solve the following system of equations:

u = 2x-y/2 (1)

v = y/2 (2)

w = z/3 (3)

Solving this system, we find x = (u + v)/2 y = 2v z = 3w

2) Let's find the new function H(u,v,w) to be integrated by substituting x, y, z in the function F(x,y,z) = x +z/3. 

3) Let's find the limits of integration of the new function i.e u.v.w:

Let's isolate x, y, z as limits of integration of the given integral. We have:

x = y/2 x = y/2 + 1

y = 0 y = 4

z = 0 z = 3

These equations represent the planes that bound the surface G in the space xyz.

From the equation x=y/2 we have 2x = y 2x-y = 0 ⇒2x-y/2 = 0 ⇒ u = 0 since u = 2x-y/2 (1)

From the equation x = y/2 + 1 we have 2x = y+2 ⇒ 2x-y = 2 ⇒ 2x-y/2 = 1  ⇒ u = 1.

Let's use the equation v = y/2:

For y = 0 we have v = 0. For y = 4 v =2 

Let's use the equation w = z/3

For z = 0 w = 0

For z =3 w = 1

Let's calculate the jacobian: