Double integrals over general regions let us add up a quantity across a curved or irregular area in the xy-plane. Instead of using constant rectangular limits, we describe the region R with functions that form its boundaries. This is essential for finding area, volume under a surface, mass of a flat plate, and average value over nonrectangular domains.
The main skill is translating a picture of a region into correct limits of integration.
Understanding Calculus: Double Integrals over General Regions
A double integral works by breaking a region into many very thin strips, then breaking each strip into tiny pieces. Each tiny piece has a small area. The function value tells how much of a quantity belongs to that small area.
Adding the contributions gives a total that becomes exact in the limit. When the inside integral is with respect to y, imagine a vertical strip. For one fixed x value, the strip starts on the lower boundary and ends on the upper boundary.
Its height can change as x moves across the region. When the inside integral is with respect to x, imagine horizontal strips instead. Their left and right edges can change with y.
A sketch is not optional for most general-region problems. Draw every boundary curve, mark where curves meet, and shade only the required area. Intersection points determine the outside limits.
For example, a line and a parabola may enclose a region only between two crossing points. Find those points before writing any integral. Then test a sample strip inside the shaded area.
A vertical strip should pass through the region in one continuous segment if it is to be described by one pair of vertical bounds. Its bottom curve must truly stay below its top curve throughout the chosen outside interval.
Some regions cannot be handled with one integral in a particular direction. A circle is a common example. Vertical strips may work smoothly, while horizontal strips can require different boundary formulas above and below a central line.
Other shapes have a boundary that changes role. A curve may be the right edge for low y values but not for high y values. In that case, split the region into simpler pieces.
Add the integrals for those pieces. Splitting is not a mistake or a last resort. It is the accurate way to describe a shape when one set of limits would otherwise include points outside the region.
Changing the order can make calculation much easier. One order may produce a complicated inner integral, while the other gives a simple polynomial or cancels a difficult expression. The geometry must be checked again from scratch when switching order.
Do not simply swap the symbols in the limits. Real uses include finding the mass of a thin metal part whose density varies by position, estimating rainfall over an uneven field, and calculating total heat across a surface. Good habits are to label the curves, state which direction the strips run, and verify that every point counted lies inside the shaded region exactly once.
A quick mental check of the units helps too. Integrating a density per unit area should produce a mass.
Key Facts
- Area of a region R: A = ∫∫_R 1 dA
- Volume under z = f(x,y) over R: V = ∫∫_R f(x,y) dA
- Type I region: R = {(x,y): a ≤ x ≤ b, g1(x) ≤ y ≤ g2(x)}
- Type I integral: ∫∫_R f(x,y) dA = ∫_a^b ∫_{g1(x)}^{g2(x)} f(x,y) dy dx
- Type II region: R = {(x,y): c ≤ y ≤ d, h1(y) ≤ x ≤ h2(y)}
- Reversing order means describing the same R with the other variable on the outside, not changing the region.
Vocabulary
- Double integral
- A double integral adds the values of a function over a two-dimensional region.
- Region of integration
- The region of integration is the set of points in the plane over which the double integral is evaluated.
- Type I region
- A Type I region is bounded vertically between a lower curve y = g1(x) and an upper curve y = g2(x).
- Type II region
- A Type II region is bounded horizontally between a left curve x = h1(y) and a right curve x = h2(y).
- Order of integration
- The order of integration tells which variable is integrated first and which variable sets the outer limits.
Common Mistakes to Avoid
- Using rectangular limits for a curved region is wrong because general regions usually need variable inner limits that follow the boundary curves.
- Putting the upper and lower curves in the wrong order is wrong because the inner limits must move from the lower boundary to the upper boundary for dy, or from left to right for dx.
- Reversing the order without redrawing the region is wrong because the new limits must describe the same set of points from the other direction.
- Forgetting to split a region when one formula does not cover the whole boundary is wrong because some regions require two or more integrals after changing order.
Practice Questions
- 1 Set up and evaluate ∫∫_R 1 dA for the region R bounded by y = x^2, y = 4, and x = 0 in the first quadrant.
- 2 Reverse the order of integration and evaluate ∫_0^2 ∫_{x^2}^4 3 dy dx.
- 3 A region is bounded by y = x and y = x^2. Explain whether it is easier to integrate first with respect to y or with respect to x, and justify your choice using the boundary curves.