Study Notes: Double Integrals and Change of Variables in Multivariable Calculus

Double Integrals and Change of Variables

Introduction to Double Integrals

Double integrals are used to compute the volume under a surface over a region in the plane. They are a fundamental concept in multivariable calculus, especially for applications in physics and engineering.

• Definition: The double integral of a function f(x, y) over a region D is denoted as $\iint_D f(x, y)\,dx\,dy$.

• Interpretation: Represents the accumulation of f(x, y) over the area D.

• Order of Integration: The order (dx then dy, or dy then dx) can be chosen based on the region's boundaries.

• Example: $\iint_D x^2 + y^2\,dx\,dy$ over a rectangular region.

Evaluating Double Integrals

To evaluate a double integral, set up the limits of integration according to the region D.

• Rectangular Regions: If D is a rectangle, limits are constants.

• General Regions: Limits may depend on x or y.

• Example: $\int_{a}^{b} \int_{c}^{d} f(x, y)\,dy\,dx$

Change of Variables in Double Integrals

Sometimes, it is easier to evaluate a double integral by changing variables, such as switching to polar coordinates or using a transformation.

• Transformation: Let x = x(u, v), y = y(u, v).

• Jacobian Determinant: The area scaling factor is given by the Jacobian $J = \left| \frac{\partial(x, y)}{\partial(u, v)} \right|$.

• Formula: $\iint_D f(x, y)\,dx\,dy = \iint_{D'} f(x(u, v), y(u, v)) |J|\,du\,dv$

• Example: Changing to polar coordinates: $x = r\cos\theta$, $y = r\sin\theta$, $J = r$.

Double Integrals in Polar Coordinates

Polar coordinates are useful for regions that are circular or have radial symmetry.

• Conversion: $x = r\cos\theta$, $y = r\sin\theta$

• Jacobian: $J = r$

• Formula: $\iint_D f(x, y)\,dx\,dy = \int_{\alpha}^{\beta} \int_{a}^{b} f(r\cos\theta, r\sin\theta) r\,dr\,d\theta$

• Example: $\iint_D x^2 + y^2\,dx\,dy$ over a disk of radius R becomes $\int_{0}^{2\pi} \int_{0}^{R} r^2 r\,dr\,d\theta$

Graphical Representation of Regions

Regions of integration are often shown on the xy-plane, with boundaries marked. Changing variables can transform these regions into rectangles or other simple shapes in the new variables.

• Rectangular Region: $D = \{ (x, y) \mid a \leq x \leq b, c \leq y \leq d \}$

• Transformed Region: $D' = \{ (u, v) \mid u_1 \leq u \leq u_2, v_1 \leq v \leq v_2 \}$

• Example: The images show rectangles and their transformation under a change of variables.

Sample Problems and Solutions

The file contains various questions involving double integrals, change of variables, and graphical regions. Here are representative examples:

• Compute: $\iint_D x^2 + y^2\,dx\,dy$ over a rectangular region.

• Change of Variables: Given $u = x + y$, $v = x - y$, find the Jacobian and set up the new integral.

• Polar Coordinates: Evaluate $\iint_D f(x, y)\,dx\,dy$ over a disk using polar coordinates.

• Graphical Region: Sketch the region D and its image under the transformation.

Key Formulas and Concepts

• Double Integral: $\iint_D f(x, y)\,dx\,dy$

• Change of Variables: $\iint_D f(x, y)\,dx\,dy = \iint_{D'} f(x(u, v), y(u, v)) |J|\,du\,dv$

• Jacobian Determinant: $J = \left| \frac{\partial(x, y)}{\partial(u, v)} \right|$

• Polar Coordinates: $x = r\cos\theta$, $y = r\sin\theta$, $J = r$

Table: Comparison of Coordinate Systems

System	Variables	Jacobian	Typical Region
Cartesian	x, y	1	Rectangle, general
Polar	r, θ	r	Circle, disk
Other (u, v)	u, v	|J|	Transformed region

Summary

Double integrals are essential for calculating areas, volumes, and other quantities over regions in the plane. Changing variables, especially to polar coordinates, simplifies integration over symmetric regions. Understanding the graphical representation of regions and the use of the Jacobian determinant is crucial for mastering these techniques.