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This double integral calculator evaluates \(\iint f(x,y)\,dA\) step by step. Enter the function and the limits for the inner and outer integral. The limits can be numbers or expressions, like \(y\) from 0 to \(x\). You get the exact answer, a decimal, and a numerical check.
How to Use It
- Type the function of x and y, for example
x yorx^2 + y^2. - Enter the inner limits (for y). They may use x, like
0tox. - Enter the outer limits (for x). These must be numbers.
- Press Solve.
The calculator integrates in the order \(dy\,dx\): first with respect to y, then x. To switch the order, swap the variable names in your function.
What Is a Double Integral?
A single integral adds up tiny strips to get an area. A double integral adds up tiny boxes over a region to get a volume under a surface \(z = f(x,y)\). If \(f(x,y) = 1\), the double integral just gives the area of the region.
We solve it as an iterated integral: do the inside integral first, treating the other variable as a constant, then do the outside one.
\[\int_a^b\!\!\int_{g_1(x)}^{g_2(x)} f(x,y)\,dy\,dx\]
Worked Examples
Example 1: \(\int_0^1\!\int_0^x xy\,dy\,dx\)
Inner: treat x as a constant. \(\int_0^x xy\,dy = x\cdot\frac{y^2}{2}\Big|_0^x = \frac{x^3}{2}\).
Outer: \(\int_0^1 \frac{x^3}{2}\,dx = \frac{x^4}{8}\Big|_0^1 = \frac18\).
Example 2: \(\int_0^1\!\int_0^2 (x^2+y^2)\,dy\,dx\)
Inner: \(\int_0^2 (x^2+y^2)\,dy = 2x^2 + \frac83\).
Outer: \(\int_0^1 \left(2x^2 + \frac83\right)dx = \frac23 + \frac83 = \frac{10}{3}\).
Example 3: \(\int_0^1\!\int_0^1 e^{x+y}\,dy\,dx\)
Since \(e^{x+y} = e^xe^y\), it splits into \(\left(\int_0^1 e^x\,dx\right)^2 = (e-1)^2 \approx 2.952\).
Tips for Setting Up Limits
- Draw the region first. Sketch the curves that bound it.
- Inner limits can depend on the outer variable. Outer limits must be constants.
- Rectangle region (\(a \le x \le b\), \(c \le y \le d\)): all limits are numbers, and the order does not matter (Fubini’s theorem).
- Triangle or curved region: one pair of limits uses the other variable.
Common Mistakes
- Putting a variable in the outer limits. The outer integral must end as a number.
- Integrating the wrong variable first. \(dy\,dx\) means y first.
- Forgetting the other variable is a constant. In \(\int xy\,dy\), x just comes along: \(\frac{xy^2}{2}\).
Learn the Math Behind the Double Integral Calculator
A double integral adds up a function over a region in the plane, giving volume under a surface or the total of a quantity spread over an area.
Frequently Asked Questions
Can I do polar double integrals?
Yes, by converting first. Replace \(x = r\cos\theta\), \(y = r\sin\theta\), and multiply by \(r\). Then type the function using y for r and x for θ, or any letters you like.
What about three variables?
Use the triple integral calculator.
How is the answer checked?
The calculator also computes the integral with Simpson’s rule (a numerical method). If both answers match, you see ✓ Verified.