How to use this calculator
Choose the order of integration, then enter the outer limits as numbers and the inner limits as functions of the outer variable (constants for a rectangle). In polar form the r in dA = r dr dθ is added for you.
Formulas
Type I∬ f dA = ∫ab∫g₁(x)g₂(x) f dy dx
Polar∬ f dA = ∫αβ∫r₁(θ)r₂(θ) f(r cos θ, r sin θ) r dr dθ
Averagefavg = (1/A)∬ f dA
Centroidx̄ = (1/A)∬ x dA, ȳ = (1/A)∬ y dA
Worked example
Stewart 15.2 Example 5: ∬ xy dA over the region between y = x − 1 and y² = 2x + 6
- The region is easiest as type II: y from −2 to 4, x from y²/2 − 3 to y + 1.
- Inner: ∫ xy dx = (y/2)[(y + 1)² − (y²/2 − 3)²].
- Outer: ∫₋₂⁴ of that dy = 36.
- Enter f = x*y, dx dy, y from −2 to 4, x from y^2/2 − 3 to y + 1.
Frequently asked questions
How do I choose the order?
Pick the order whose inner limits are single formulas. If a vertical strip crosses different boundary curves in different parts of the region, use horizontal strips or split the region.
When should I switch to polar coordinates?
When the region is a disk, ring or sector, or f contains x² + y². Remember the extra factor r.
Is the answer exact?
It is a numerical value accurate to about 7 significant figures, from adaptive quadrature on both integrals.