How to use it
- Type the integrand, for example
x*y^2. - Enter the x limits and the y limits of the rectangle. They can be expressions such as
pi. - Read the integral, the area and the average value.
How it works
For each x the inner integral ∫ f(x, y) dy from c to d is computed with adaptive Simpson’s rule, which splits the interval until the estimate stops changing. The outer integral does the same over x from a to b, calling the inner one at every sample point. The inner integrals use a tighter tolerance so their small errors do not add up in the outer one.
Worked example
∬ x·y² dA over 0 ≤ x ≤ 2, 0 ≤ y ≤ 3. Because the integrand is a product, the integral is ∫₀² x dx · ∫₀³ y² dy = 2 · 9 = 18. The calculator shows 18, the area 2 · 3 = 6 and the average value 18 / 6 = 3.
Tips and common mistakes
- Functions that blow up inside the rectangle, such as 1/x with x = 0 on the edge, give an error rather than a wrong number.
- For a volume under a surface, f must be the height above the xy-plane; parts below the plane count as negative.
- Products such as sin(x)·cos(y) separate into two single integrals, which is a quick way to check the result.