Double Integral Over Rectangle- Formula and Examples
What Is a Double Integral Over a Rectangle?
A double integral over a rectangle is exactly what it sounds like: integrating a function of two variables over a rectangular region. Instead of finding the area under a curve (single integral), you're finding the volume under a surface.
The region of integration is always rectangular: a ≤ x ≤ b and c ≤ y ≤ d. That's the constraint. If your region isn't a rectangle, this specific technique won't work—you'll need to adjust the limits.
The notation looks like this:
∬R f(x, y) dA
Where R is the rectangle and dA represents the differential area element. You can write dA as dx dy or dy dx—the order matters for computation, but Fubini's Theorem says the result is the same if the function is continuous.
The Formula
For a function f(x, y) over rectangle R = [a, b] × [c, d]:
∬R f(x, y) dA = ∫ab ∫cd f(x, y) dy dx
Or equivalently:
∬R f(x, y) dA = ∫cd ∫ab f(x, y) dx dy
The inner integral runs first, then the outer. Evaluate the inner one, plug in limits, then handle the outer one.
Iterated Integrals: The Inner-Outer Structure
Think of it as two separate integrals chained together. The inner integral treats all x-terms as constants. The outer integral then finishes the job.
Example structure when integrating in order dy dx:
- Integrate with respect to y first (x stays fixed)
- Substitute the y-limits (c and d)
- Integrate the resulting expression with respect to x
- Substitute the x-limits (a and b)
Getting Started: A Step-by-Step Process
Step 1: Identify the Region
Write down your bounds clearly. If the problem states integrate f(x, y) over 0 ≤ x ≤ 2, 1 ≤ y ≤ 3, then those are your limits.
Step 2: Choose Integration Order
Pick dy dx or dx dy. Some functions are easier one way. If f(x, y) contains ex², integrate with respect to y first—integrating ex² with respect to x is a dead end.
Step 3: Set Up the Iterated Integral
Write the full expression with limits. Don't skip this step. Many mistakes happen from writing limits incorrectly.
Step 4: Evaluate the Inner Integral
Hold the outer variable constant. Treat everything with the inner variable as your integration variable.
Step 5: Evaluate the Outer Integral
Take the result from Step 4 and integrate with respect to the remaining variable.
Example 1: Basic Calculation
Evaluate ∬R (x + 2y) dA where R: 0 ≤ x ≤ 1, 1 ≤ y ≤ 2
Setting up the iterated integral (integrating y first, then x):
∫01 ∫12 (x + 2y) dy dx
Inner integral (treat x as constant):
∫12 (x + 2y) dy = [xy + y²]12 = (2x + 4) - (x + 1) = x + 3
Outer integral:
∫01 (x + 3) dx = [½x² + 3x]01 = ½ + 3 = 3.5
The answer is 7/2.
Example 2: Product Function
Evaluate ∬R xy² dA where R: 1 ≤ x ≤ 3, 0 ≤ y ≤ 2
∫13 ∫02 xy² dy dx
Inner integral:
∫02 xy² dy = x · [y³/3]02 = x · (8/3) = 8x/3
Outer integral:
∫13 8x/3 dx = 8/3 · [x²/2]13 = 8/3 · (9/2 - 1/2) = 8/3 · 4 = 32/3
The answer is 32/3.
Example 3: Swapping Integration Order
Evaluate ∫02 ∫y2 ex² dx dy
Here the inner integral is dx. But ex² has no elementary antiderivative. Swap the order.
The region: y ≤ x ≤ 2, 0 ≤ y ≤ 2. This means 0 ≤ y ≤ x and 0 ≤ x ≤ 2.
Rewriting:
∫02 ∫0x ex² dy dx
Inner integral:
∫0x ex² dy = ex² · y|0x = xex²
Outer integral (use u-substitution: u = x², du = 2x dx):
∫02 xex² dx = ½ ∫04 eu du = ½ [eu]04 = ½(e⁴ - 1)
The answer is (e⁴ - 1)/2.
Volume Interpretation
The double integral ∬R f(x, y) dA gives the volume between the surface z = f(x, y) and the xy-plane over the rectangle R—if f(x, y) ≥ 0.
For negative values, the integral gives signed volume. Parts below the xy-plane subtract from the total.
Comparing Integration Orders
| Situation | Best Order | Why |
|---|---|---|
| f contains ex² or ey² | Integrate the other variable first | No elementary antiderivative for x² in exponent |
| Region given as y-bounds in terms of x | dy dx | Limits already set up this way |
| Region given as x-bounds in terms of y | dx dy | Easier to use given bounds directly |
| Constant integrand f(x, y) = 1 | Either | Result is just the area of the rectangle |
Common Mistakes
- Forgetting to substitute limits on the inner integral before moving to the outer. Always evaluate and simplify completely.
- Mixing up the order of integration with the limits. If inner bounds depend on the outer variable, you must integrate the inner one first.
- Treating the outer variable as constant inside the inner integral. Only the inner variable changes during that step.
- Algebra errors when expanding. Take it slow on the arithmetic.
Quick Reference
For rectangle [a, b] × [c, d] and function f(x, y):
- Volume = ∬R f(x, y) dA
- Average value = (1/area) · ∬R f(x, y) dA
- Area of R = (b-a)(d-c)
That's the core of double integrals over rectangles. Practice swapping order and always check if one order leads to an impossible integral before you start.