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.
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.