Double Integral Calculator
Calculate $\int \int f(x,y) \, dA$ step-by-step
$$ \int_{0}^{1} \int_{-\sqrt{1-x^2}}^{\sqrt{1-x^2}} (x^2 + y^2) \, dy \, dx $$
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Final Result
2D Region of Integration (Domain D)
Step-by-Step Solution
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By Prof. David Anderson
Professor of Mathematics | 20+ Years Experience
“In Calculus III, standard integration is just the beginning. The real challenge with Double Integrals isn’t the calculus itself—it’s the geometry. Setting up the bounds ($dy dx$ vs $dx dy$) and converting to Polar Coordinates are where most students struggle. I designed this Double Integral Calculator to visualize the region $D$ and handle the heavy lifting of Iterated Integration for you.”
The Ultimate Guide to Double Integrals: Cartesian, Polar, and Changing Order
How to Use a Double Integral Calculator for Volume, Area, and Mass
A Double Integral ($\iint_D f(x,y) dA$) extends the concept of integration to functions of two variables. It calculates the Volume under the surface $z = f(x, y)$ over a specific region $D$ in the xy-plane.
Solving these requires mastering two critical skills: setting up Iterated Integrals (Type I vs Type II regions) and knowing when to use Polar Coordinates. Our Double Integration Solver handles both Cartesian and Polar forms automatically.
1. Double Integral Notation & Fubini’s Theorem
We evaluate a double integral by performing two single integrations in sequence. This is called an Iterated Integral.
Fubini’s Theorem (Cartesian)
$$ \iint_D f(x,y) \, dA = \int_{a}^{b} \left[ \int_{g_1(x)}^{g_2(x)} f(x,y) \, dy \right] dx $$
Inner: Integrate wrt $y$ (treat $x$ constant).
Outer: Integrate wrt $x$ (constants only).
Outer: Integrate wrt $x$ (constants only).
2. Choosing Your Coordinate System
The secret to solving double integrals easily is picking the right coordinate system for your region $D$.
Method A
Cartesian (Rectangular)
Best for regions bounded by vertical/horizontal lines or functions like $y=x^2$.
$dA = dy \, dx$ OR $dx \, dy$
Use Type I (vertical arrows) or Type II (horizontal arrows).
Method B
Polar Coordinates
Best for circles, rings, or cardioids.
$dA = r \, dr \, d\theta$
Jacobian Correction: Don’t forget to multiply by $r$!
3. Step-by-Step Integration Protocol
Manual calculation requires strict adherence to the “Partial Integration” rule. Here is the protocol used by our Double Integral Calculator.
Step 1
Set the Limits
Sketch the region $D$. Determine if you should integrate $y$ first (vertical strips) or $x$ first (horizontal strips). The outer limits MUST be constants.
Step 2
Inner Integral
Integrate the function with respect to the inner variable. Treat the other variable as a constant coefficient (Partial Integration).
Step 3
Outer Integral
Plug in the bounds from Step 2. You should now have a function of only one variable. Integrate normally to get the final numeric value.
4. Master Class: Worked Examples
Type A: Cartesian Volume
Type I Region
Evaluate $\int_0^2 \int_0^x (x + 2y) \, dy \, dx$.
1. Inner Integral (wrt y)
$$ \int_0^x (x + 2y) dy = [xy + y^2]_0^x $$
(Treat x as constant)
2. Plug Bounds (y=0 to y=x)
$$ (x(x) + x^2) – (0) = 2x^2 $$
3. Outer Integral (wrt x)
$$ \int_0^2 2x^2 dx = [\frac{2}{3}x^3]_0^2 = \frac{16}{3} $$
Type B: Polar Conversion
Circular Region
Find the volume of $z = e^{-x^2-y^2}$ over the unit circle $D: x^2+y^2 \le 1$.
Convert to Polar: $x^2+y^2 = r^2$, $dA = r dr d\theta$.
1. Setup Integral
$$ \int_0^{2\pi} \int_0^1 e^{-r^2} \cdot r \, dr \, d\theta $$
2. Inner Integral (u-sub: $u=-r^2$)
$$ \int_0^1 re^{-r^2} dr = [-\frac{1}{2}e^{-r^2}]_0^1 = \frac{1}{2}(1 – e^{-1}) $$
3. Outer Integral (wrt $\theta$)
$$ \int_0^{2\pi} \text{Constant} \, d\theta = 2\pi \cdot \frac{1}{2}(1 – \frac{1}{e}) = \pi(1 – \frac{1}{e}) $$
5. Advanced: Changing Order of Integration
Sometimes, an integral like $\int \int e^{y^2} dx dy$ is impossible to solve in the given order. You must Change the Order of Integration using Fubini’s Theorem. This usually simplifies the problem significantly.
Step 1
Sketch the Region
Draw the region $D$ defined by the original bounds (e.g., $x$ goes from $y$ to $1$).
Step 2
Switch Directions
If original was Type II (horizontal arrows), switch to Type I (vertical arrows) and find new bounds for $y$ in terms of $x$.
6. Professor’s FAQ
When should I use Polar Coordinates?
Use Polar Coordinates whenever the region $D$ is circular (circle, ring, sector) OR if the integrand contains terms like $x^2 + y^2$. This simplifies the integration significantly.
What is the Jacobian?
The Jacobian is the “correction factor” when changing coordinate systems. For Polar Coordinates, the Jacobian is $r$. That’s why $dA$ becomes $r \, dr \, d\theta$, not just $dr \, d\theta$.
Does order matter (dy dx vs dx dy)?
Mathematically, if the function is continuous, Fubini’s Theorem says the result is the same. However, computationally, one order is often much easier (or even possible) than the other. Our Iterated Integral Calculator helps you test both.
References & Further Reading
- Stewart, J. (2020). Calculus: Early Transcendentals (9th ed.). Cengage Learning. (Chapter 15: Multiple Integrals).
- Paul’s Online Math Notes. “Double Integrals.” Lamar University.
- MIT OpenCourseWare. “Multivariable Calculus.” 18.02.
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