Integrating Over a Rectangle- Double Integral Tutorial
What Is a Double Integral Over a Rectangle?
A double integral over a rectangle extends single-variable integration into two dimensions. You're summing up values of a function f(x, y) across a two-dimensional region that happens to be rectangular. That's it. No magic here.
If you visualize a surface z = f(x, y) hovering over a rectangle R in the xy-plane, the double integral ∬R f(x, y) dA gives you the volume between that surface and the rectangle. When f(x, y) is positive, the integral measures volume. When f is negative, you're subtracting volume below the xy-plane.
The notation dA represents a tiny piece of area. You can write it as dx dy or dy dx—the order doesn't matter for rectangular regions.
The Geometric Meaning
Think of slicing the surface with vertical planes parallel to the axes. You get thin columns with base area dA and height f(x, y). The double integral adds up all these columns.
For a constant function f(x, y) = c over a rectangle with width a and height b, the integral is simply c × a × b. This is the volume of a rectangular box with height c.
When the Function Is Always Positive
Double integrals are straightforward: the result equals the volume under the surface. A positive function above a positive region gives positive volume.
When the Function Has Negative Values
The integral subtracts volume below the xy-plane. The total integral can be positive, negative, or zero depending on what cancels out. If you need the actual geometric volume, you split the region where f changes sign and integrate each piece separately.
How to Evaluate Double Integrals
You evaluate double integrals by converting them into iterated integrals. This means you integrate with respect to one variable at a time while treating the other as constant.
Fubini's Theorem for Rectangles
If f is continuous on R = [a, b] × [c, d], then:
∬R f(x, y) dA = ∫ab ∫cd f(x, y) dy dx = ∫cd ∫ab f(x, y) dx dy
The inner integral runs first, then the outer. You can swap the order—the result stays the same for rectangular regions.
Working Inside Out
Start with the inner integral. Treat everything except the integration variable as constant. Solve it, then integrate the result with respect to the outer variable.
Practical Examples
Example 1: Basic Calculation
Evaluate ∬R (x + 2y) dA where R = [0, 2] × [0, 3].
Using the order dy dx:
∫02 ∫03 (x + 2y) dy dx
Inner integral (treat x as constant):
∫03 (x + 2y) dy = [xy + y²]03 = 3x + 9
Outer integral:
∫02 (3x + 9) dx = [3⁄2x² + 9x]02 = 6 + 18 = 24
Example 2: xy dx dy Order
Evaluate ∬R xy² dA over R = [1, 3] × [0, 2].
Using the order dx dy:
∫02 ∫13 xy² dx dy
Inner integral:
∫13 xy² dx = y² [x²/2]13 = y² (9/2 - 1/2) = 4y²
Outer integral:
∫02 4y² dy = [4y³/3]02 = 32/3
Example 3: Non-Separable Function
Evaluate ∬R ex+y dA where R = [0, 1] × [0, 1].
∫01 ∫01 ex+y dy dx
ex factors out of the inner integral since it's constant with respect to y:
Inner: ∫01 exey dy = ex[ey]01 = ex(e - 1)
Outer: (e - 1)∫01 ex dx = (e - 1)[ex]01 = (e - 1)²
Properties of Double Integrals
Double integrals over rectangles follow predictable rules:
- Addition: ∬R [f + g] dA = ∬R f dA + ∬R g dA
- Scalar multiplication: ∬R cf dA = c ∬R f dA for any constant c
- Dominance: If f ≤ g on R, then ∬R f dA ≤ ∬R g dA
- Rectangle containment: If R₁ ⊆ R₂, then ∬R₁ f dA ≤ ∬R₂ f dA when f ≥ 0
Order of Integration: Which to Pick?
Both orders give the same answer for rectangles. But sometimes one order is much easier. Here's how to choose:
- Pick the order that matches how f separates into x and y parts
- If f(x, y) = g(x)h(y), integrate each factor with its own variable
- If one variable has complicated limits, make it the outer integral
Swapping the Order
When you swap from dy dx to dx dy, the limits must change accordingly. Draw the region. The original inner variable's range becomes the outer variable's range, and vice versa.
Average Value of a Function
The average value of f over R is:
favg = (1/Area of R) ∬R f(x, y) dA
For a rectangle with area A = (b-a)(d-c), the average value is the integral divided by the area. This is the 2D analog of the single-variable formula.
Common Mistakes to Avoid
- Forgetting to treat the outer variable as constant when doing the inner integral
- Mixing up the limits—each variable needs its own bounds
- Skipping the geometric interpretation when the problem asks for volume
- Confusing the double integral with two separate single integrals
Quick Reference Table
| Topic | Key Point |
|---|---|
| Notation | ∬R f dA = ∬R f(x,y) dx dy |
| Geometric meaning | Volume under z = f(x,y) over R |
| Fubini's Theorem | Swap integration order freely for rectangles |
| Evaluation method | Iterated integrals, work inside out |
| Average value | Integral divided by rectangle area |
Getting Started: Your First Double Integral
Step 1: Identify the region R. For a rectangle, this means finding [a, b] × [c, d].
Step 2: Choose an integration order. Try dy dx first—usually simpler for beginners.
Step 3: Write the iterated integral with limits filled in.
Step 4: Evaluate the inner integral, treating the outer variable as constant.
Step 5: Plug the result into the outer integral and evaluate.
Step 6: Check your work. Can you verify using the opposite order?
When Double Integrals Over Rectangles Show Up
You'll encounter these in probability (joint distributions over rectangular support), physics (center of mass calculations, moments of inertia), and engineering (flux through rectangular surfaces). The rectangular case is the foundation—once you master this, general regions become an extension, not a new concept.
Practice with constant functions first to build intuition, then move to separable functions, then everything else. The pattern is always the same: convert to iterated form, integrate twice, done.