01 Key Concepts
Double Integrals
Integral over a region R of f(x,y) dA computes the volume under a surface f(x,y) above a 2D region R.
Iterated Integrals
A double integral is usually computed as two single integrals done one after another — integrate with respect to one variable first, treating the other as constant, then integrate the result.
Order of Integration
dx dy or dy dx — the order depends on how the region's boundaries are described; sometimes one order is much easier than the other.
Setting Up Bounds
Inner bounds may depend on the outer variable (e.g., y from 0 to x), while outer bounds are always constants.
Applications
Double integrals compute volumes under surfaces, areas of 2D regions, and total mass when density varies across a region.
02 Key Formulas
- Integral over R of f(x,y) dA = Integral (outer bounds) of [Integral (inner bounds) of f(x,y) d(inner var)] d(outer var)
03 Solved Examples
- Integrate with respect to y first, treating x as constant: Integral from 0 to 3 of x*y dy = x*y^2/2 evaluated from 0 to 3 = x*(9/2).
- Now integrate with respect to x: Integral from 0 to 2 of (9x/2) dx = (9/2)*(x^2/2) evaluated from 0 to 2 = (9/2)*2.
- Integrate with respect to y: Integral from 0 to 2 of 1 dy = 2.
- Integrate with respect to x: Integral from 0 to 1 of 2 dx = 2*1.
- Integrate with respect to y first: Integral from 0 to 1 of (x+y) dy = xy + y^2/2 evaluated from 0 to1 = x + 1/2.
- Integrate with respect to x: Integral from 0 to 1 of (x + 1/2) dx = x^2/2 + x/2 evaluated from 0 to 1 = 1/2 + 1/2.
04 Practice Questions
📄 Multiple Integrals — Downloadable Worksheet
10 questions with a full answer key. Grab the PDF to print, or try the interactive version in your browser.