4.4 Change of Variables in Multiple Integrals

Coordinate Transformations

Cartesian to Polar Transformation

When an integration region is circular or the integrand contains $x^2 + y^2$, polar coordinates almost always simplify the problem. The transformation maps each point using:

$x = r\cos(\theta), \quad y = r\sin(\theta)$

The area element changes too. In Cartesian coordinates you write $dA = dx\,dy$, but in polar coordinates this becomes:

$dA = r\,dr\,d\theta$

That extra factor of $r$ is the Jacobian for this transformation. It accounts for the fact that small "patches" of area get larger as you move away from the origin.

Setting up a polar integral:

1. Sketch the region and identify the radial and angular bounds
2. Replace $x$ and $y$ in the integrand using the transformation equations above
3. Replace $dA$ with $r\,dr\,d\theta$
4. Write the limits: radial bounds on $r$ (inner integral) and angular bounds on $\theta$ (outer integral), or vice versa depending on the region

For a full disk of radius $R$, the limits are $0 \leq r \leq R$ and $0 \leq \theta \leq 2\pi$. A semicircle or wedge-shaped region just narrows the $\theta$ range.

Triple Integrals in Cylindrical and Spherical Coordinates

Cylindrical coordinates extend polar into 3D by keeping the $z$-axis unchanged:

$x = r\cos(\theta), \quad y = r\sin(\theta), \quad z = z$

The volume element is $dV = r\,dr\,d\theta\,dz$. Use cylindrical coordinates when the region has circular symmetry around the $z$-axis but isn't necessarily spherical (think cylinders, cones, or paraboloids).

Spherical coordinates use a radial distance $\rho$ from the origin and two angles:

$x = \rho\sin(\phi)\cos(\theta), \quad y = \rho\sin(\phi)\sin(\theta), \quad z = \rho\cos(\phi)$

Here $\phi$ is the angle measured down from the positive $z$-axis ($0 \leq \phi \leq \pi$) and $\theta$ is the usual angle in the $xy$-plane. The volume element is:

$dV = \rho^2\sin(\phi)\,d\rho\,d\phi\,d\theta$

The $\rho^2\sin(\phi)$ factor is the spherical Jacobian. Use spherical coordinates when the region is a sphere, a cone, or any shape defined naturally by distance from the origin.

To set up a triple integral in either system:

1. Identify which coordinate system matches the geometry of the region
2. Substitute the coordinate transformation into the integrand
3. Replace $dV$ with the appropriate volume element (including the Jacobian factor)
4. Determine the limits for each variable from the geometry of the region

Change of Variables and Simplification

Jacobian Determinant for Variable Changes

Polar, cylindrical, and spherical coordinates are specific cases of a more general idea: you can define any change of variables $(x, y) = T(u, v)$ and integrate in the new system. The Jacobian determinant tells you how the transformation stretches or compresses area (or volume) at each point.

For a 2D transformation, the Jacobian is the determinant of the matrix of partial derivatives:

$J = \frac{\partial(x,y)}{\partial(u,v)} = \begin{vmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{vmatrix}$

For 3D, you get a $3 \times 3$ determinant:

$J = \frac{\partial(x,y,z)}{\partial(u,v,w)} = \begin{vmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} & \frac{\partial x}{\partial w} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} & \frac{\partial y}{\partial w} \\ \frac{\partial z}{\partial u} & \frac{\partial z}{\partial v} & \frac{\partial z}{\partial w} \end{vmatrix}$

The change of variables formula for double integrals is:

$\iint_R f(x,y)\,dx\,dy = \iint_S f(x(u,v),\, y(u,v))\,|J|\,du\,dv$

and for triple integrals:

$\iiint_R f(x,y,z)\,dx\,dy\,dz = \iiint_S f(x(u,v,w),\, y(u,v,w),\, z(u,v,w))\,|J|\,du\,dv\,dw$

You always take the absolute value of $J$. The region $S$ in the new coordinates corresponds to the original region $R$.

Steps for a general change of variables:

1. Define the transformation: write $x, y$ (and $z$) in terms of the new variables
2. Compute all the partial derivatives and build the Jacobian matrix
3. Evaluate the determinant and take its absolute value
4. Substitute the transformation into the integrand
5. Transform the integration limits from the old region to the new one
6. Write the new integral and evaluate

As a quick check: for polar coordinates, computing the 2D Jacobian gives $|J| = r$, which matches the $r\,dr\,d\theta$ you already know.

Choosing the Right Coordinate System

Picking the best coordinate system is half the battle. Here's how to decide:

• Circular or radial symmetry in 2D (disk, annulus, cardioid): use polar
• Circular cross-sections with a vertical axis in 3D (cylinder, cone, paraboloid): use cylindrical
• Spherical symmetry (sphere, hemisphere, regions defined by distance from origin): use spherical
• Rectangular regions with straight-line boundaries: stick with Cartesian
• Integrand contains $x^2 + y^2$: polar or cylindrical will likely simplify it
• Integrand contains $x^2 + y^2 + z^2$: spherical is a strong candidate

When none of the standard systems fit, consider a custom substitution. For example, if the region is bounded by lines like $y = 2x$ and $y = x + 3$, you might set $u = y - 2x$ and $v = y - x$ to turn the region into a rectangle in $uv$-space. The trade-off is that you'll need to compute the Jacobian yourself, so weigh the complexity of the Jacobian against how much simpler the limits and integrand become.