Introduction
A double integral adds up a function over a flat region. You do it in two steps: integrate the inside variable first, then the outside one. This double integral calculator does both steps for you and shows the work.
Type your function, like x*y^2. Pick definite and add the four limits, or pick indefinite and skip them. Then press Calculate. You get an exact answer, a decimal answer, and a clear step-by-step solution. For single-variable problems, the Integral Calculator is the simpler tool to reach for.
You can also switch the order of integration, work in Cartesian or polar coordinates, and use your own variable names. The graph tab draws a 3D surface of your function and a 2D picture of the region you are integrating over, so you can see what the math means. If you want to explore surfaces on their own, try the 3D Graphing Calculator. There is a math keyboard for symbols like π and √, sample problems to try, and a "Check My Answer" box that tells you if your own answer is right.
How to use our Double Integral Calculator
Type in your function, pick your two variables, and add your limits. The calculator shows the exact answer, a decimal answer, step-by-step work, and graphs of the surface and the region.
Coordinate system: Pick Cartesian (x, y), Polar (r, θ), or General. This sets the default variable names for you. For polar, remember to put the extra r in your function yourself.
Outer variable: Type the letter that gets integrated last, like x. Use one symbol only.
Inner variable: Type the letter that gets integrated first, like y. It must be different from the outer variable.
Integrand: Type the function only, such as x*y^2 or sin(x)*cos(y). Do not type dx or dy. Use ^ for powers and * for times. Trig values can be checked quickly with the Trig Calculator.
Show Keyboard: Click this to tap math keys like √, π, sin, and fractions right into the box you last clicked.
Camera button: Upload a picture of your integral, then type what you see into the box and apply it to the integrand.
Limits of integration: Choose Definite to add four bounds and get a number. Choose Indefinite to skip bounds and get an antiderivative plus C.
Inner lower bound: Type where the inner variable starts. This one may use the outer variable, like 0 or 1-x.
Inner upper bound: Type where the inner variable stops. It may also use the outer variable, like x^2.
Outer lower bound: Type the number where the outer variable starts. It must be a constant, like 1 or pi/2.
Outer upper bound: Type the number where the outer variable stops. It must be a constant too.
Preferred integration method: Leave it on Auto-detect, or pick a method like substitution, parts, or partial fractions if your class needs it. If a rational integrand needs polynomial division first, the Long Division Calculator and Synthetic Division Calculator can help you set it up.
Calculate: Click to solve. You get the exact answer, the decimal answer, the steps, and the plots.
Switch Integration Order: Click to flip dy dx into dx dy. The calculator rewrites the bounds when it can.
Clear All: Click to reset every field back to the starting example.
Check My Answer: Type your own answer, like 85/8 or pi/3, and click Verify to see if it matches. The Fraction Calculator is handy if you need to tidy up your fraction first.
What Is a Double Integral?
A double integral adds up a function over a flat region instead of just along a line. You write it as ∫∫ f(x, y) dy dx. If f(x, y) is a height, the double integral gives the volume between the surface and the region below it. If f(x, y) = 1, it gives the area of the region. For simple shapes you can skip the calculus and use the Area Calculator or the Volume Calculator instead.
How Double Integrals Work
You solve a double integral in two rounds, one variable at a time. This idea is called Fubini's theorem.
- Inner integral first. Integrate with respect to the inner variable (like y). Treat the other variable (x) as a plain number.
- Plug in the inner limits. Put in the top bound, then subtract the bottom bound. The answer is now a function of x only.
- Outer integral next. Integrate that result with respect to x.
- Plug in the outer limits. Top minus bottom. You now have one number.
Each round undoes a derivative, so if you are checking your antiderivative by differentiating it back, the Derivative Calculator is the fastest way to confirm it.
Limits of Integration
The order of the limits matters a lot:
- Inner limits can be numbers or curves that use the outer variable, like y = 0 to y = x².
- Outer limits must always be plain constants, like x = 1 to x = 2.
- The inner limits can never contain the inner variable.
If both inner limits are numbers, the region is a rectangle. If an inner limit uses the outer variable, the region has curved or slanted edges. To find where two boundary curves cross, the System of Equations Calculator and the Quadratic Formula Calculator come in handy.
Switching the Order of Integration
Sometimes an integral is hard in one order and easy in the other. You can swap dy dx to dx dy, but you must redraw the region and rewrite the bounds. Over a rectangle, the swap is simple. Over a curved region, you must solve the boundary curve for the other variable — the Solve for X Calculator can do that rearranging for you.
Polar Coordinates
Circles, rings, and pie slices are much easier in polar form. There you use r (distance from the center) and θ (angle). One rule you cannot skip: the extra factor r, called the Jacobian. So dA becomes r dr dθ. Forgetting the r is the most common polar mistake. For plain circular regions, the Circle Area Calculator and the Arc Length Calculator give quick sanity checks.
Common Uses
- Volume under a surface — compare with the Cylinder Volume Calculator for standard solids
- Area of a flat region
- Mass of a thin plate with changing density
- Average value of a function over a region, similar in spirit to the Average Calculator
- Center of mass and moments in physics and engineering, which feed into the Moment of Inertia Calculator
- Probability with two random variables — see the Probability Calculator and the Normal Distribution Calculator
Common Methods You May Need
Each round is just a single integral, so the usual tools work: the power rule, u-substitution, integration by parts, trig identities, partial fractions, and completing the square. Related helpers include the Exponent Calculator for power rules, the Log Calculator for logarithmic antiderivatives, the Factoring Calculator for partial fractions, and the Limit Calculator when a bound approaches a tricky value.
Mistakes to Watch For
- Treating the outer variable as a variable during the inner step. It acts like a constant.
- Matching the wrong limits to the wrong variable.
- Using a variable in the outer limits.
- Typing dx or dy inside the function itself.
- Dropping the r in polar coordinates.
- Swapping the order without changing the bounds.
- Rounding too early — keep extra digits and use the Sig Fig Calculator only at the very end.