Double Integral Calculator

A double integral adds up f(x, y) over a region R. You integrate the inner variable first, treating the other as a constant, then integrate the result. For example, \(\int_{0}^{2}\int_{0}^{x} xy^{2}\,dy\,dx = \int_{0}^{2} \frac{x^{4}}{3}\,dx = \frac{32}{15}\).

How to use the double integral calculator

1. Enter f(x, y).
2. Choose the order: dy dx means y is integrated first (inner), then x.
3. Enter the inner limits (they may use the outer variable) and the outer limits (numbers).
4. For polar regions choose Polar; the calculator includes the extra factor r automatically.
5. Press Integrate to see the steps, the exact value when one exists, and the region.

How double integrals are evaluated

By Fubini’s theorem, a double integral is computed as an iterated integral:

\(\iint_{R} f(x,y)\,dA = \int_{a}^{b}\left[\int_{g_{1}(x)}^{g_{2}(x)} f(x,y)\,dy\right]dx\)

1. Integrate the inside with respect to y, holding x constant.
2. Substitute the inner limits.
3. Integrate the resulting function of x between the outer limits.

Worked example: region between two curves

Find the area between \(y = x^{2}\) and \(y = \sqrt{x}\) for \(0 \le x \le 1\):

\(\int_{0}^{1}\int_{x^{2}}^{\sqrt{x}} 1\,dy\,dx = \int_{0}^{1}\left(\sqrt{x} – x^{2}\right)dx = \left[\tfrac{2}{3}x^{3/2} – \tfrac{1}{3}x^{3}\right]_{0}^{1} = \tfrac{1}{3}\)

Polar coordinates

For disks, rings and sectors use \(x = r\cos\theta,\ y = r\sin\theta\) and \(dA = r\,dr\,d\theta\):

\(\int_{0}^{2\pi}\int_{0}^{2} r^{2}\cdot r\,dr\,d\theta = \int_{0}^{2\pi} 4\,d\theta = 8\pi\)

Forgetting the factor r is the most common polar mistake.

Changing the order of integration

Some integrals are easy in one order and impossible in the other. \(\int_{0}^{1}\int_{y}^{1} e^{x^{2}}\,dx\,dy\) has no elementary inner antiderivative, but the same region described as \(0 \le y \le x,\ 0 \le x \le 1\) gives \(\int_{0}^{1} x e^{x^{2}}\,dx = \frac{e-1}{2}\). When no exact form is found, the calculator still gives an accurate decimal.

Double integrals practice quiz: 10 questions

Test yourself with 10 multiple-choice questions. Choose an answer and press Submit to see if you are right and why. Your score appears at the end.

Frequently asked questions

How do you solve a double integral?

Integrate the inner integral with respect to its variable while treating the other variable as a constant, substitute the limits, then integrate the result with respect to the outer variable.

Does the order of integration matter?

For continuous functions the value is the same in either order (Fubini’s theorem), but the limits change and one order can be much easier.

Why is there an extra r in polar double integrals?

A small polar region has area approximately r·dr·dθ, so dA = r dr dθ.

What if there is no antiderivative?

The calculator then evaluates the integral numerically with adaptive Gauss–Kronrod quadrature and shows a decimal accurate to about 8 digits.

Can the inner limits be functions?

Yes. Inner limits may use the outer variable, such as y from x² to √x. The outer limits must be numbers.

Sources

• Paul’s Online Math Notes: Double Integrals
• Paul’s Online Math Notes: Double Integrals in Polar Coordinates
• OpenStax Calculus Volume 3: Multiple Integration

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