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.
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. 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.
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.
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.
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.
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.
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, and the Solve for X Calculator handles that rearranging.
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.
Common Uses
- Volume under a surface
- Area of a flat region
- Mass of a thin plate with changing density
- Average value of a function over a region
- Center of mass and moments in physics and engineering
- Probability with two random variables
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.
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 round only at the very end.