Change the order of integration

1. Why Change Order

Some iterated integrals cannot be done in given order (inner integral non-elementary, e.g. $∫ e^{y²} dy$ ) or region description is messy. Changing order ( $dy\,dx ↔ dx\,dy$ ) often makes it elementary or simpler. Fubini guarantees equality when $f$ continuous (absolute integrability).

2. Method — Sketch and Re-describe $D$

Given $I = ∫_{x=a}^{b} ∫_{y=g_1(x)}^{g_2(x)} f(x,y) dy dx$ :

Sketch $D$ bounded by $y=g_1(x), y=g_2(x), x=a, x=b$ .
Describe $D$ the other way: $x = h_1(y) to h_2(y)$ , $y∈[c,d]$ .
Write $I = ∫_{y=c}^{d} ∫_{x=h_1(y)}^{h_2(y)} f(x,y) dx dy$ .

For $D$ that is not both x- and y-simple, split into pieces in new order.

3. Worked Examples

(a) I = ∫_{0}^{1}∫_{y}^{1} e^{y²}?? Actually classic: ∫_{0}^{1}∫_{x}^{1} e^{y²} dy dx cannot do inner e^{y²}?
    Reverse: D: x 0→1, y x→1 is triangle y≥x. Other way: y 0→1, x 0→y:
    I = ∫_{0}^{1}∫_{0}^{y} e^{y²} dx dy = ∫_{0}^{1} y e^{y²} dy = (e-1)/2 ✓

(b) I = ∫_{0}^{a}∫_{y}^{a} x²/√(x²+y²) ?? Often polar needed but order change may help.

(c) I = ∫_{0}^{1}∫_{y}^{1} sin(y²) dy dx? No sin? Actually ∫_{0}^{1}∫_{x}^{1} sin(y²) dy dx = ?
    D as above: = ∫_{0}^{1}∫_{0}^{y} sin(y²) dx dy = ∫_{0}^{1} y sin(y²) dy = (1-cos1)/2 ✓

(d) I = ∫_{0}^{2}∫_{y/2}^{1} e^{x²} dx dy? Provided? Another: ∫_{0}^{1}∫_{0}^{√(1-x²)} f? Change.

(e) D bounded by y= x² and y= x: ∫_{0}^{1}∫_{x²}^{x} f dy dx ; other order: y 0→1, x y→√y
    ⇒ ∫_{0}^{1}∫_{y}^{√y} f dx dy ✓

(f) I = ∫_{0}^{4}∫_{y/2}^{2} e^{x²} ?? D: y=0→4, x= y/2 →2 is same as x 0→2, y 0→2x
    So I = ∫_{0}^{2}∫_{0}^{2x} e^{x²}? Wait integrand? Example shows region re-description.

(g) Sometimes need two pieces: D = 0≤x≤1, 0≤y≤x plus 1≤x≤2, 0≤y≤2-x? Combined gives triangle with apex.

4. Dirichlet and Other Tricks

If integrand separates as $f(x)g(y)$ over rectangle, $∬ = (∫f)(∫g)$ — no order needed. But change is essential when inner antiderivative is not elementary.

5. Exam Points

Given $∫_{a}^{b}∫_{g_1(x)}^{g_2(x)} f dy dx$ , change to $dx\,dy$ with new limits — 5-7 marks, very frequent.
Evaluate originally non-elementary $∫∫ e^{y²}$ or $sin y² / y$ via change — 7 marks, very frequent.
Sketch region and split if needed — 5 marks, frequent.

Change the order of integration