Questions: Double Integrals in Cartesian Coordinates

Question 1

You are computing ∬_R f(x,y) dA over the region bounded by y = x² (below) and y = x (above) for 0 ≤ x ≤ 1. A student sets up ∫₀¹ ∫₀¹ f(x,y) dy dx. What is wrong?

• A. The outer limits should run from 0 to 1 in y, not x, since y is the inner variable
• B. The inner limits should be functions of x (from x² to x), not the constants 0 to 1
• C. Double integrals over non-rectangular regions can only be evaluated in polar coordinates
• D. Nothing is wrong — Fubini's theorem allows constant limits regardless of the region's shape

Answer: (B)

The error is treating a non-rectangular region as if it were a rectangle. For a fixed x in [0,1], y doesn't range over all of [0,1] — it only ranges from the lower boundary x² up to the upper boundary x. The correct setup is ∫₀¹ ∫_{x²}^{x} f(x,y) dy dx. Option D is the classic mistake: Fubini's theorem guarantees that switching integration order gives the same answer for well-behaved f, but it does NOT allow you to ignore the actual shape of the region.

Question 2

When should you prefer slicing a region horizontally (fixing y and integrating x from a left boundary to a right boundary) over slicing vertically?

• A. Always — horizontal slicing is the standard convention in Cartesian double integrals
• B. Only when f(x,y) depends solely on y and not on x
• C. When the horizontal boundaries are simpler to express as functions of y than the vertical boundaries as functions of x
• D. Only when the region is symmetric about the y-axis

Answer: (C)

The choice of slicing direction is a strategic one based on which description of the region's boundary gives cleaner limits. For the region between y = x and y = x², vertical slices give simple limits (y from x² to x). But a different region — say, bounded by x = y² and x = y — is more naturally described with horizontal slices. Always sketch the region first and ask: which boundary curves are easier to express as functions of the outer variable? That's the direction to slice.

Question 3

For any region R, the double integral ∬_R 1 dA equals the area of R.

• T. True
• F. False

Answer: True

When f(x,y) = 1, the double integral sums up infinitesimal area elements dA over the entire region, giving the total area. This is a useful check: if you compute a 'volume' with f = 1 and get a negative number, or if your area is implausibly large, the limits are wrong. More generally, if f represents surface density, ∬_R f dA gives total mass — the double integral is a summation over the region, and f = 1 is just the constant-density (uniform) case.

Question 4

To reverse the order of integration in a double integral over a non-rectangular region, you can simply swap the inner and outer bounds without re-examining the region.

• T. True
• F. False

Answer: False

Reversing integration order on a non-rectangular region requires re-describing the region's boundaries from scratch for the new slicing direction. The old limits cannot simply be swapped — they were derived for a specific slicing direction and are only valid for that direction. You must sketch the region again, determine the new outer variable's range, and re-derive the inner variable's bounds as functions of the new outer variable. Swapping without this step produces wrong limits.

Question 5

Why does the inner integral in a double integral over a non-rectangular region have limits that are functions of the outer variable, rather than constants?

Model answer: Because for each fixed value of the outer variable, the range of the inner variable depends on where the region's boundaries are at that particular cross-section. Think of slicing the region into thin strips: each strip at position x spans a different y-range (from the lower boundary curve g₁(x) to the upper boundary curve g₂(x)). Constants would only work if every strip had the same height — which is true only for rectangles.

The variable limits are what distinguish integrating over a shaped region from integrating over a rectangle. They encode the geometry of the region directly into the integral. Getting them right requires understanding the region's shape, which is why sketching before setting up is essential — the limits are a description of the region's boundaries, and that description changes depending on which direction you slice.