Change of Variables in Double and Triple Integrals

CHANGE OF VARIABLES IN DOUBLE AND TRIPLE INTEGRALS

We have seen in section 5.1.5 the change of variables in double integral into polar coordinates and in section 5.2.1, the change of variables in triple integral into cylindrical coordinates and spherical polar coordinates.

We shall now see the change of variables other than the above transformations.

1. Suppose x = g(u, v) and y = h(u, v) be the transformation and g, h have continuous partial derivatives, then

Here we assume the transformation is one-one.

2. Suppose x = f1(u, v, w), y = f2(u, v, w), z = f3(u, v, w) be the transformation where f1, f2, f3 have continuous partial derivatives and the transformation is one-one, then dx dy dz = |J|du dv dw

WORKED EXAMPLES

Example 1 

Evaluate

Solution 

⸫ u varies from 0 to 1 and v varies from 0 to 2

We shall sketch the regions R in the xy-plane and S in the uv-plane.

We have x = u + v and y = 2v

Example 2 

Evaluate where R is the region in the xy-plane bounded by the x-axis and the parables y2 = 4(x + 1), y2 = 4(1 − x) above the x-axis, using the transformation x = u2- v2, y = 2uv

Solution 

We shall sketch the region R in the xy-plane and the transformed region S in the uv-plane. Given R is bounded by the x-axis y = 0, the parabolas y2 = 4(x + 1) and y2 = 4(1 − x)

For the parabola y2 = 4(x + 1)

The vertex is (-1, 0). The axis is the x-axis y = 0 and towards the positive x-axis. 

For the parabola y2 = 4(1 − x)

The vertex is (1, 0). The axis is the x-axis y = 0 and towards the negative x-axis. The transformation is x = u2 - v2, and y = 2uv. Since the region is above the x-axis y ≥ 0.

⸫ u varies from 0 to 1 and v varies from 0 to 1

So, the transformed region S is a square in the uv-plane.

Where J is the Jacobian of transformation.

Example 3 

Evaluate where R is the parallelogram in the xy-plane with vertices (1, 0), (3, 1), (2, 2), (0, 1), using the transformation u = x + y, v = x -2y.

Solution 

Where R is the parallelogram with vertices A(1, 0), B(3, 1), C(2, 2), D(0, 1). We shall sketch the region R in the xy-plane and the transformed region S in the uv-plane.

The transformation is u = x + y and v = x – 2y

We find the images of the vertices

Adding (3) and (4), we get

Adding (5) and (6), we get

The transformed region S is a square in the uv-plane.

Where u varies from 1 to 4 and v varies from -2 to 1

But

Here f(x, y) = (x + y)2 = u2

Example 4 

Evaluate where R is the triangular region with vertices (0,0), (2, 1) and (1, 2), using the transformation x = 2u + v, y = u + 2v

Solution 

Where R is the triangular region in the xy-plane with vertices 0(0, 0), A(2, 1), B(1, 2). We shall sketch the region R in the xy-plane and the transformed region S in the uv-plane. The transformation is x = 2u + v and y = u + 2v.

We shall how sketch R in the xy-plane and S in the uv-plane.

Where u varies from u = 0 to u = 1 – v and varies from v = 0 to v = 1.

We have x = 2u + v and y = u + 2v

Example 5 

Evaluate where the region R is enclosed by the lines x + y = 1, x + y =3, x – y = 1 and x – y = -1 using the transformation u = x + y, v = x - y.

Solution 

Let where R is the region in the xy-plane bounded by the line x + y = 1, x + y = 3, x − y = 1, x − y = −1. 

The transformation is u = x + y, v = x - y

⸫ The side x + y = 1 ⇒ u = 1, x – y = 1 ⇒ v = 1

x + y = 3 ⇒ u = 3, x – y = -1 ⇒ v = -1

We shall sketch the region R in the xy-plane and S in the uv-plane

R is a square in the xy-plane and S is square in the uv-plane u varies from 1 to 3 and v varies from -1 to 1

Example 6 

Evaluate where R is the triangle bounded by x = 0, y = 0, x + y = 1 using the transformation u = x + y and y = uv.

Solution 

Let where R is the triangle bounded by

x = 0, y = 0, x + y = 1.

The transformation is u = x + y, y = uv.

Now u = x + y ⇒ u = x + uv ⇒ x = u - uv

= u(1 - v)

and y = uv

When y = 0, uv = 0 ⇒ u = 0 or v = 0

When x = 0, u(1 - v) = 0 ⇒ u = 0 or v = 1

When x + y = 1, u = 1

⸫ S is bounded by u = 0, u = 1, v = 0, v = 1. So, S is a square in the uv-plane.

⸫ u varies from u = 0 to u = 1; v varies from v = 0 to v = 1

Example 7 

Find the volume of the region R in space enclosed by the ellipsoid using the transformation x = au, y = bv  z = cw.

Solution 

The volume of the region R is given by I =

The transformation is x = au, y = bv, z = cw

Where J is the Jacobian of transformation.

Where V is the volume of the sphere u2 + v2 + w2 = 1

Example 8

Solution 

We shall find the transformed region S in the uvw-space.

When y = 0, v = 0

When y = 4, v = 2

When z = 0, w = 0

When z = 3, w = 1

⸫ S in the uvw-space is bounded by the planes u = 0, u = 1, v = 0, v = 2, w = 0, w = 1.

Which is a rectangular parallelepiped.

⸫ u varies from u = 0 to v = 1

v varies from v = 0 to v = 2

w varies from w = 0 to w = 1

Where J is the Jacobian of transformation.

Example 9 

Evaluate where R is bounded by the surfaces x = 1, x = 3, z = y, z = y + 1, xy = 2, xy = 4. The transformation is u = x, v = 2 - y, w = xy.

Solution 

Where the region R in the xyz-space is bounded by the surfaces x = 1, x = 3, z = y, z = y + 1, xy = 2, xy = 4.

The transformation is u = x ⇒ x = 4 

v = z - y ⇒ z = v + y

Now we shall find the region in the uvw-space.

⸫ So, S is a rectangular parallel piped.

Where J is the Jacobian of transformation

EXERCISE

ANSWERS TO EXERCISE