MAST20029 · Engineering Mathematics
Double Integrals and Change of Order
Double integrals ∬_R f dA are set up over a general region by slicing it into vertical strips (x outer) or horizontal strips (y outer). The chapter's signature exam skill — and the classic mid-semester-test question — is changing the order of integration: re-describing the region so that an otherwise impossible inner integral (like ∫ e^(x³) dx) becomes doable. Marks come from sketching the region correctly and re-reading its limits, not from clever integration.
What this chapter covers
- 011. The double integral ∬_R f dA as a volume / accumulation over a planar region
- 022. Type-I regions: a ≤ x ≤ b with g₁(x) ≤ y ≤ g₂(x) (vertical strips, x outer)
- 033. Type-II regions: c ≤ y ≤ d with h₁(y) ≤ x ≤ h₂(y) (horizontal strips, y outer)
- 044. Reading the limits straight off a sketch of the region
- 055. Changing the order of integration by re-describing the same region
- 066. When to swap: the inner antiderivative has no closed form one way (e.g. ∫ e^(x³) dx)
- 077. Substitution to finish the swapped integral (u = x³, etc.)
- 088. Sketching as the safety net — never flip limits mechanically
Change the order of integration to make it doable
- 1 markRead the region from the given order: 0 ≤ y ≤ 1 and √y ≤ x ≤ 1. The bound x ≥ √y means y ≤ x², so the region is bounded above by y = x² with x running from 0 to 1.
- 1 markRe-describe with x outer (type-I): 0 ≤ x ≤ 1 and 0 ≤ y ≤ x². Sketch confirms the same triangular-ish region under y = x².
- 1 markSwap the order: ∫₀¹ ∫₀^(x²) e^(x³) dy dx = ∫₀¹ e^(x³) · x² dx, since e^(x³) is constant in y and the inner length is x².
- 1 markSubstitute u = x³, du = 3x² dx, so x² dx = du/3: ∫₀¹ e^(x³) x² dx = (1/3) ∫₀¹ e^u du = (1/3)(e − 1).
- 1 markState the answer: (1/3)(e − 1).
Key terms
- Double integral ∬_R f dA
- The integral of a function of two variables over a planar region R, evaluated as an iterated integral once R is described by limits in a chosen order.
- Type-I region
- A region described as a ≤ x ≤ b with g₁(x) ≤ y ≤ g₂(x): vertical strips with y as the inner variable (x outer).
- Type-II region
- A region described as c ≤ y ≤ d with h₁(y) ≤ x ≤ h₂(y): horizontal strips with x as the inner variable (y outer).
- Change of order
- Re-describing the SAME region with the other variable inner, so the bounds swap. Used when the inner antiderivative does not exist in one order but does in the other.
- Iterated integral
- The evaluation of a double integral as two single integrals done from the inside out; the inner limits may depend on the outer variable but never the reverse.
- Region sketch
- A drawing of R from its boundary curves — the reliable way to read off correct limits in either order and to avoid mechanical limit-flipping errors.
Double Integrals and Change of Order FAQ
When should I change the order of integration?
Two triggers. First, when the inner integral has no elementary antiderivative in the given order (the classic examples are e^(x²), e^(x³), sin(x²), and (sin x)/x) but becomes a simple substitution after swapping. Second, when one order forces a region to be split into several pieces but the other order covers it in one. Sketch the region, re-describe it, then swap.
How do I avoid mistakes when swapping the limits?
Sketch the region first. Identify the boundary curves, then read the new limits straight off the picture: the new outer variable runs over its full numeric range, and the new inner limits are the curves that bound the region in that direction. Never just exchange the symbols in the limits — that is the most common and most costly error.
What is the difference between type-I and type-II descriptions?
They describe the same region in opposite slicing directions. Type-I uses vertical strips (x outer, a ≤ x ≤ b, with y between two functions of x); type-II uses horizontal strips (y outer, c ≤ y ≤ d, with x between two functions of y). Changing the order means switching between these two descriptions of one region.
Is change of order examined?
Yes — it is a signature mid-semester-test and exam skill in the vector-calculus block. The marks are for correctly sketching and re-describing the region; the integration after the swap is usually a short substitution. Show the sketch, the new limits, and one line of substitution.
Exam move
Treat every awkward double integral as a region problem, not an integration problem. Step one is always to sketch R from its boundary curves; step two is to re-describe it in the other order by reading the limits off the sketch; only then do you integrate. Build a reflex for the swap triggers — an inner antiderivative that does not exist (e^(x²), e^(x³), sin(x²), sin x / x) almost always means you should change the order. Practise the full ritual end to end (sketch → new limits → substitution → answer) on a handful of past MST-style problems, and write the region description explicitly so the marker sees you understood the geometry.
