What is a Double Integral?

A double integral extends the concept of a single integral to functions of two variables. Written as \(\iint_R f(x,y)\,dA\), it represents the accumulation of a function over a two-dimensional region \(R\). Double integrals are fundamental in calculus for computing areas, volumes under surfaces, mass of laminas, centers of mass, moments of inertia, and probabilities in continuous probability distributions.

The double integral of \(f(x,y)\) over region \(R\) gives the signed volume between the surface \(z = f(x,y)\) and the xy-plane. When \(f(x,y) = 1\), the double integral simply computes the area of region \(R\).

How to Compute a Double Integral

To evaluate a double integral, you typically convert it to an iterated integral using Fubini's theorem, which states that under certain continuity conditions:

The computation process follows these steps:

• Step 1: Identify the region of integration and determine appropriate bounds
• Step 2: Choose the order of integration (dy dx or dx dy) based on the region's geometry
• Step 3: Integrate with respect to the inner variable, treating the outer variable as constant
• Step 4: Evaluate the inner integral at its bounds to obtain a function of the outer variable
• Step 5: Integrate the resulting expression with respect to the outer variable
• Step 6: Evaluate at the outer bounds to obtain the final numerical result

Iterated Integrals: dy dx vs dx dy

The order of integration matters when dealing with non-rectangular regions. Understanding when to use each order is crucial for successful evaluation.

Type I Region (dy dx order)

A Type I region is bounded by vertical lines \(x = a\) and \(x = b\), with the top and bottom defined by functions of \(x\): \(y = g_1(x)\) and \(y = g_2(x)\). The integral is:

Use this when the region is easier to describe with \(x\) varying over a fixed interval and \(y\) depending on \(x\).

Type II Region (dx dy order)

A Type II region is bounded by horizontal lines \(y = c\) and \(y = d\), with left and right boundaries defined by functions of \(y\): \(x = h_1(y)\) and \(x = h_2(y)\). The integral is:

Use this when the region is easier to describe with \(y\) varying over a fixed interval and \(x\) depending on \(y\).

Choosing the Right Order

Sometimes changing the order of integration can simplify the calculation significantly. If one order leads to an integral you cannot evaluate symbolically, try switching to the other order. The result should be the same by Fubini's theorem, but the difficulty of computation may differ dramatically.

Double Integral Formula

The formal definition of a double integral uses a Riemann sum limit. For a region \(R\) partitioned into small rectangles of area \(\Delta A_i\), we have:

where \((x_i^*, y_i^*)\) is a sample point in the \(i\)-th rectangle. This definition motivates numerical integration methods, which approximate the integral by evaluating the function at a finite grid of points.

For practical computation, we use the iterated integral formulas mentioned above. The key formulas are:

• Type I (vertical slices): \(\displaystyle\int_a^b \int_{g_1(x)}^{g_2(x)} f(x,y)\,dy\,dx\)
• Type II (horizontal slices): \(\displaystyle\int_c^d \int_{h_1(y)}^{h_2(y)} f(x,y)\,dx\,dy\)
• Rectangular region: \(\displaystyle\int_a^b \int_c^d f(x,y)\,dy\,dx\) where the bounds are all constants

Examples of Double Integrals

Example 1: Simple Polynomial over Rectangle

Problem: Evaluate \(\displaystyle\int_0^1 \int_0^2 (x+y)\,dy\,dx\)

Solution:

Inner integral: \(\displaystyle\int_0^2 (x+y)\,dy = \left[xy + \frac{y^2}{2}\right]_0^2 = 2x + 2\)

Outer integral: \(\displaystyle\int_0^1 (2x+2)\,dx = \left[x^2 + 2x\right]_0^1 = 1 + 2 = 3\)

Answer: 3

Example 2: Type I Region

Problem: Evaluate \(\displaystyle\int_0^1 \int_0^x xy\,dy\,dx\)

Solution: The region is bounded by \(x = 0\), \(x = 1\), \(y = 0\), and \(y = x\) (a triangular region).

Inner integral: \(\displaystyle\int_0^x xy\,dy = x\left[\frac{y^2}{2}\right]_0^x = \frac{x^3}{2}\)

Outer integral: \(\displaystyle\int_0^1 \frac{x^3}{2}\,dx = \frac{1}{2}\left[\frac{x^4}{4}\right]_0^1 = \frac{1}{8}\)

Answer: 1/8 = 0.125

Example 3: Trigonometric Function

Problem: Evaluate \(\displaystyle\int_0^\pi \int_0^{\pi/2} \sin(x)\cos(y)\,dy\,dx\)

Solution:

Inner integral: \(\displaystyle\int_0^{\pi/2} \sin(x)\cos(y)\,dy = \sin(x)[\sin(y)]_0^{\pi/2} = \sin(x)\)

Outer integral: \(\displaystyle\int_0^\pi \sin(x)\,dx = [-\cos(x)]_0^\pi = -(-1-1) = 2\)

Answer: 2

Example 4: Type I with Quadratic Bounds

Problem: Evaluate \(\displaystyle\int_0^1 \int_{x}^{1} (x^2+y^2)\,dy\,dx\)

Solution: The region is bounded by \(y = x\) (below) and \(y = 1\) (above), for \(0 \leq x \leq 1\).

Inner integral: \(\displaystyle\int_x^1 (x^2+y^2)\,dy = \left[x^2y + \frac{y^3}{3}\right]_x^1 = x^2 + \frac{1}{3} - x^3 - \frac{x^3}{3} = x^2 - \frac{4x^3}{3} + \frac{1}{3}\)

Outer integral: Integrate and evaluate from 0 to 1 to get the final answer (calculation requires algebraic simplification).

Common Mistakes

• Incorrect bounds order: Mixing up which bounds correspond to which variable. Always check that inner bounds match the inner variable and can depend on the outer variable.
• Forgetting to treat the outer variable as constant: When integrating with respect to \(y\), treat all \(x\) terms as constants, and vice versa.
• Incorrect limit evaluation: After integrating, carefully substitute the upper bound minus the lower bound. Sign errors are common here.
• Wrong region setup: Sketching the region before setting up bounds helps avoid errors. Ensure your bounds actually describe the intended region.
• Forgetting absolute value: If computing area and the function is negative over part of the region, you may need \(\iint |f(x,y)|\,dA\).
• Not checking order of integration: If symbolic integration fails, try reversing the order. Some integrals are much easier in one order than the other.
• Arithmetic errors in numeric methods: Ensure your grid resolution is sufficient. Check convergence by comparing results at different resolutions.

Frequently Asked Questions

Learn more: Paul's Online Math Notes - Double Integrals | OpenStax Calculus - Double Integrals