Double Integral Calculator: Integrate f(x, y) over a Region with Variable y Limits
Double integrals over a region bounded by curves.
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How this calculator works
Type f(x, y).
Enter the x limits and the y limits, which may be expressions in x.
The calculator gives the integral, the area and the average value.
Formulas used
- \int_{a}^{b}\int_{g_1(x)}^{g_2(x)}f(x,y)\,dy\,dx
- A = \int_{a}^{b}\left(g_2(x) - g_1(x)\right)dx
- \bar f = \frac{1}{A}\iint f\,dA
Worked example
With the default numbers in the calculator above:
- Value of the double integral: 2
- Area of the region: 2
- Average value of the function over the region: 1
Change the numbers in the calculator to see the full working, every step, for your own case.
A second worked example
A different case, with these numbers: f(x, y) x^2+y^2; x from 0; x to 1; y from (a number or an expression in x) 0; y to (a number or an expression in x) 1.
- Value of the double integral: 0.6666667
- Area of the region: 1
- Average value of the function over the region: 0.6666667
Common questions
How do I find the area with a double integral?
Integrate the function 1 over the region.
What if my region is described with x depending on y?
Swap the names x and y in the expression and the limits so the outer variable is the one with constant limits.
How accurate is it?
It uses 20-point Gauss-Legendre integration, which is exact for polynomials and accurate to about 6 figures for smooth functions. Edges with a square-root shape are a little less accurate.
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Got a question about this? Ask an engineering question and get a plain-language answer with references.
Further reading: Any multivariable calculus text (double integrals).
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